Three princes and three princesses are ready to marry.
Everybody independently chooses a partner without telling anyone.
What is the probability that everybody chooses a person who chooses her/him?
Assuming that boys choose girls and girls choose boys.
There are 8 weights with mass of 1, 2, 3, 4, 5, 6, 7, and 8 kilograms (kg). Is it possible to place them in the corners of a cube so that the center of their mass coincides with the center of the cube?
The question of I. Akulich, Kvant, 1993, 9-10.
You have been invited to a game of dice (with no fee to watch). The dice seem fair and standard 6-sided dice.
You throw two dice and pick the greatest value showing between the two dice.
You pay $10 per game. You nominate a number between 1 and 6 inclusive.
If you nominate 1, and 1 wins, you get $350.
If you nominate 2, and 2 wins, you get $118.
If you nominate 3, and 3 wins, you get $72.
If you nominate 4, and 4 wins, you get $50.
If you nominate 5, and 5 wins, you get $41.
If you nominate 6, and 6 wins, you get $32.
Which number should you nominate?
In the ancient kingdom of Diceon it has been decreed that the standard 6-sided dice that have been used for millennia are offensive to the gods. From now on, in any game involving dice, the numbers 3 and 4 are no longer acceptable outcomes. If such an invalid number is thrown, the player must utter a simple incantation and take that throw again.
However, the Diceon-gods are magnanimous, and take pity on a player who initially scores only one. In this case the player gets another throw. If and only if another one is thrown, the player's score is increased to 2. (Otherwise, the player's score remains at 1.)
What is the probability of scoring 2?
by Leslie Green
In a rural community only the Elders and the vicar are allowed to wear hats as they enter the church. However, nobody can wear a hat in the church, so the Elders leave their hats with an altar boy, who gives them a ticket for their hat. This enables them to collect their hat when they leave.
There are four Elders, but the illiterate altar boy, not only mixes up the vicar's hat with those of the Elders, he also ignores the ticket numbers, returning the hats to the Elders randomly.
What is the probability that exactly one of the Elders is given a hat which is not his?
A dice game is played using standard 6-sided dice, one being red and the other blue. The red player pays $1.00 per game whereas the blue player pays some other fixed amount X in dollars. The blue player can only win a game if the blue die value exceeds the red die value. There is no possibility of a draw. The winner takes both payments.
You have been asked to play blue all night.
What is the greatest value of X that you should be willing to pay?
An old 4-digit padlock has become corroded with time such that only two of the 4 tumblers actually have a locking function. To be clear, the non-functioning tumblers can be in any position and this does not prevent the lock from opening. The other two tumblers work normally. There is no 'feel' to the tumblers which tells you which ones are faulty or which ones are working.
Using an optimal strategy, what is the minimum number of trials necessary to guarantee opening the lock?
Guess the rule of writing numbers in the squares.
Replace dots by the numbers.
Which is the sum of numbers in the bottom line?
On a remote island local scientists create a new alphabet.
They checked all existing words (1 million) and found that
there are twice as many words with the first letter of the alphabet than with the second letter,
there are twice as many words with the second letter than with the third letter,
there are twice as many words with the third letter than with the fourth letter,
and on so on.
Estimate the largest possible number of letters in the alphabet.
There are four prisoners numbered 1 to 4 (inclusive) with big numbers on their chests. Every Christmas the warden allows them to be released only if all four of them find their numbers in the secret room after at most two guesses. The prisoners have all year to decide on a strategy, but on the day they are not allowed to communicate at all.
The guard takes each prisoner into the secret room where there are four numbered opaque baskets, each covering a large numbered tile. These numbered tiles are arranged in some random fashion. The guard has been chosen as somebody of impeccable character (honest and fair minded).
The prisoner picks one basket to lift. If the prisoner has found his number he leaves, and the guard notes the success. If the result was a failure the prisoner gets one more chance. He leaves in any case, but the guard notes the success or failure. The guard then returns the baskets to their original positions so later prisoners gain no information by basket alignment. The prisoner always leaves by a different door than the one they came in.
Using the optimum strategy, what is the chance that the prisoners are released at Christmas?
Adrian and Brian are placed in two separate rooms. Adrian tosses a coin and Brian also tosses a coin. Adrian has to guess the outcome of Brian's toss and Brian guesses Adrian's. If both guesses are right, they win $12, otherwise nothing.
What do they expect to win per game on average if they choose the best strategy?
You have a two sided coin; heads on one side and tails the other.
You toss the coin 10 times and get 5 heads.
You toss the same coin 100 times and get 45 heads.
You toss the same coin 1000 times and get 410 heads.
You toss the same coin 10,000 times and get 4000 heads.
You toss the same coin 100,000 times and get 40,000 heads.
Estimate the probability of tossing a tail.
You probably had to memorize the “multiplication table” up to 10 in your primary school. The sum of all the blue numbers in the table is 3025.
What is the sum of all the numbers in the 15x15 table?
Ann starts at point A, and each minute, walks one block northward or eastward randomly choosing one of the two directions until she reaches point B.
Ben starts at point B, and each minute, walks one block southward or westward randomly choosing one of the two directions until he reaches point A.
What is the probability that Ann and Ben meet during their walks?
The problem is similar to a question of Presh Talwalkar
In a yard, 100 chicks sit in a circle.
At 12:01, 12:02, 12:03, . . . , each chick randomly pecks the chick immediately to its left or right.
At what time is the expected number of pecked chicks equal to 100?
The numbers from 1 to 16 are placed clockwise on a circle. We move around the circle clockwise erasing every other number until only one number remains.
If we start by erasing 1, what is the last remaining number on the circle?
An urn contains two red balls, and three balls of arbitrary non-red colours. Jane and Gerry take turns randomly picking a ball from the urn, with Jane going first.
Which is the probability of Jane picking the last red ball from the urn?
An urn contains fifteen balls. Gerry and Jane are going to take turns randomly drawing balls from the urn (without replacement), and whoever draws more red balls wins.
Jane draws first. She must remove all balls of a certain chosen color from the urn before the game starts.
Which balls should she remove from the urn?
Three brothers who lived in the old German city of Köln decided to equally share 9 amphorae of grain without opening them.
The first amphora contains 1 quart of grain,
the second amphora 2 quarts of grain,
. . .
the nineth amphora 9 quarts of grain.
How many different ways were there to equally share the 9 amphorae among the three brothers?
This is a problem from an old 13th century German book
A magician suggests that you choose a number from the set { 1, 2, 3, 4, 5, 6 } before throwing 3 fair dice.
If you get at least one of your chosen number then you win.
Estimate the probability of your win.
There are four candidates in an election. After 97% of the votes had been counted, the preliminary results were as follows:
Anna: 90,
Boris: 95,
Cindy: 100,
Donald: 103.
How many people still have a chance of winning the election?
A machine with unlimited capacity dispenses four tokens for $1.
Red tokens are worth 1 point, yellow tokens are worth 5 points, blue tokens are worth 10 points, and green tokens are worth 20 points. The machine dispenses the 4 tokens randomly choosing their values. We can get 4 points, 80 points, or many other different number of points for $1.
How many different numbers of points can we get for $1?
Randomly pick 118 natural numbers.
Consider their residues, modulo 117.
What is the probability that at least two of these numbers have the same modulo 117 residue?
Ann and Brian are both new to archery, and still pretty useless. They struggle to hit the target.
Brian hits the target once out of every four shots, on average.
Ann is slightly better. She hits the target once out of every three shots.
These two novices compete, shooting off arrows one at a time until somebody wins by actually hitting the target.
Since Brian is worse, he gets to go first.
What is the chance that Brian wins?
Ann hits a small target 75% of the time and Bob only 25% of the time.
After they shoot together an arrow each, only one of them hits the target.
What is the probability that it is Ann's arrow?
Consider the following experiment:
STEP 1A: Two balls, labelled 1 and 2 respectively, are placed into an urn.
STEP 1B: One ball is randomly chosen from the urn and then discarded.
STEP 2A: Two balls, labelled 3 and 4 respectively, are placed into the same urn as in step 1A.
STEP 2B: One ball is randomly chosen from the urn and then discarded.
Define PN(1) as the probability that ball 1 remains in the urn at the end of step NB.
Evaluate PN(2).
One hundred otherwise-identical consecutively numbered balls (001-100 inclusive) are placed into an urn. Balls are randomly removed from the urn, and then discarded.
What is the probability that ball 037 remains in the urn after 25 random removals?
The big symbol, which perhaps looks a bit like a capital E, is actually the Greek capital letter sigma. In scientific circles it is used as a concise way of writing a summation, as demonstrated in the example shown in the upper part of the image.
Evaluate the summation described by the big-sigma notation given in the lower part of the image.
There are four light switches, and there are four light bulbs.
Each switch connects to exactly one light bulb, and each light bulb connects to exactly one switch. Sadly the information about which switch connects to which light bulb has been lost by the incompetent electrician who did the work.
How many different wiring patterns are possible?
You throw a pair of standard fair 6-sided dice.
What is the most probable prime sum of their values?
Two dice are thrown and the score is the value of the largest number showing.
What is the average value in such a game?
In a fair game using a standard 6-sided die, the value on the top face of the die after being thrown is the amount you get paid in dollars.
If you throw a 4 you get $4.
What is the maximum amount you should pay to play this game?
You will be able to play this game as many times as you like at the same cost per game.
Your assistant throws a fair 6-sided die, but withholds the resulting number from you. You should guess the value using as few tries as possible.
Your guess consists of stating a number, and your assistant then says: "greater", "equal", or "lesser" in response.
Using an optimum sequence of guesses on each throw, what is the average number of guesses needed per throw?
A blind darts player throws a single dart at a graduated scale some distance away. The metal spike of the dart is pointy at the end, but soon becomes a 1 mm diameter. The dart is thrown hard enough to enter the scale up to the non-tapering part of the dart.
The graduations on the scale are marked with integers at 20 mm intervals. However, the actual scale lines themselves are so infeasibly thin you would need an electron microscope to see them. The darts player has an assistant so the dart goes into the scale and not into nearby people or objects. If the dart fails to hit the scale, the throw is repeated until it does hit.
What is the probability that the dart hits the infeasibly thin line?
Q' from the Q-continuum (of Star Trek) places infinitely many posts on an infinite plane, where the posts are 1 m between centres on a square grid as shown.
(Q' is pronounced Q-prime)
Q'' is then challenged to remove half of the posts in a systematic manner such that the minimum gap between posts is maximally increased.
What is the new minimum gap?
by Leslie Green
Spot the pattern to establish the value of the ratio involving the natural power m.
HINT: Do not worry too much about the funny lim value on the left. It reads as "The limit as n tends towards infinity". All it means is that as n increases, the resultant ratio gets closer and closer to the value given. In effect the error becomes negligibly small when n is sufficiently large.
by Leslie Green
Each natural number N has a position on this circle:
1 goes to 1; 2 goes to 2; 3 goes to 3
4 goes to 4; 5 goes to 1
...
Where does N go to?
We will use the C-language modulus operator for simplicity.
In these examples the modulus is 3.
1 % 3 = 1
2 % 3 = 2
3 % 3 = 0
A square tile has been placed near the middle of a large floor. It is permanently fixed in place.
As a first step two tiles are simultaneously but randomly placed next to the first tile, such that their edges are completely aligned. To be clear, the two new tiles are only required to align with the first tile.
As a second step a single tile is randomly placed, and is required to align with one edge of the existing tile placements.
What is the probability that a square shown at the right has been formed?
Everyone knows you mustn't go through the Dark Forest. You have a 1 in 4 chance of drowning in a swamp. You have a 1 in 5 chance of having your head bitten off by a flesh-eating mutant Venus Flytrap. You have a 1 in 6 chance of brushing against a "Touch-of-Death" orchid, and being slowly and painfully poisoned to death.
Brenda the Bold enters the Dark Forest, with a plan to go through it. What is the probability that she succeeds, and lives to tell the tale?
Secret Santa is a Western Christmas tradition in some workplaces in which each person anonymously receives a gift from one of the other members of the group.
All the participants put their names in a hat. Each player will draw a name in turn. If they should draw their own name they will return it to the hat and draw another. If the last person draws her/his name the draws start again from the beginning.
Alan, Betty, Carol, and Dan are to play a game of Secret Santa. They will draw in alphabetical order.
Estimate the probability that Alan and Betty exchange gifts as well as Carol and Dan.
Considering only whole numbers, N, between 1 and 20 (inclusive), how many have the property
sumOfDigits(N) = sumOfDigits(N2) ?
For example,
sumOfDigits(11) = 2
sumOfDigits(112) = sumOfDigits(121) = 4
Hence 11 does not have the required property.
Eight figure skaters participate in a competition. Each of three judges independently assigns places 1 to 8 to the skaters, where 1 is the best. The skater with the smallest sum of the places was the winner.
All the judges gave the same place to the only winner.
What is the largest (worst) possible placing the winner could have received?
In this 4 x 4 grid we make exactly two numbers per row negative (by prefixing the numbers with a minus sign), subject to the constraint that exactly two numbers per column must also be negative.
Estimate how many different variants are possible, giving the best estimate you can find.
You throw two fair dice.
What is the probability that one number is at least two bigger than the other?
You have a possibility to throw a die up to three times. You will earn the face value of the die in dollars. You have the option to stop after each throw and take the money earned (on that throw only). To be clear, you cannot take any earlier higher score.
What is the expected payoff of the game, assuming you use an optimum strategy?
We randomly pick two squares on the 2 x 2 grid.
What is the probability that the squares share a common side?
Gerry goes from the centre 0 along the green roads. He randomly chooses left or right at the road forks.
What is the probability that Gerry finishes his journey at one of the four points A, B, C, or D?
In a competition of 4 judokas, a girl fights with three others once.
All of them have the same probability to win and there are no draws.
What is the probability that some contestant will win all of their matches?
Go from A to B along the square net always to the right or up and without crossing the red diagonal.
How many different ways are there?
A bus stops in Jane's small village every 10 minutes.
Jane's wait for the bus is usually about 5 minutes on average.
Today, the first bus arrives at 8:50 instead of 8:10 and the next 5 buses arrive with an interval of 2 minutes.
How will the average waiting time change if she arrives at a random moment between 8:00 and 9:00 a.m.?
A young couple would like to have four children.
What is the probability of giving birth to three children of one sex and one of the other?
The country's statistics shows that each birth has an equal chance of being a boy or a girl.
The picture shows 13 squares whose side-lengths are 1, 2, 3, . . , 13 cm.
What fraction of the area of the outer square is green?
You wish to find an unknown whole number that is in the range from 1 to 8 inclusive. The only type of question you are allowed to ask is "Is the value greater than [N]?" The expression [N] means that you can substitute any number for [N]. As an example, you could ask "Is the value greater than 3?"
The only answer you will be given is "yes" or "no".
How many such questions do you need to ask to find the unknown number?
Three large triangles contain small blue and white triangles.
How many small white and blue triangles are there in a large triangle that has 10 blue triangles in its base?
Write the numbers 1, 2, 3, 4, 5, 6, 7 in the circles so that a number is always greater than the connected number below it.
How many different ways to fill the circles are there?
by Sandor Roka
Two standard, fair six-sided dice are rolled. The product of the two numbers rolled is calculated.
What product has more chances to be obtained?
In a class, ½ of students has a brother/sister and/or a pet.
¼ of students with a brother/sister have a pet.
⅓ of students with a pet have a brother/sister.
Only Jane and Johnny have both a sibling and a pet.
How many students are there in the class?
On a particular radio station they play a game called "show me the money". Having heard the jingle, you text in your name and one person is randomly selected. That person can either accept the £10,000 win or they shout "SHOW ME THE MONEY". This then converts the win into either £20,000 or £5,000 with equal probability.
From a purely mathematical point of view, which choice gives a greater expectation value? (Best win on average).
77 standard dice show a total score of 277.
What is the total of the 77 numbers which are hidden underneath each of the dice?
Four kids have a Christmas egg each.
They sit around a round table, close their eyes, and when the clock strikes one hour they move their eggs either to the left or to the right, absolutely randomly choosing one of the two sides.
What is the probability that everybody has an egg after the blind transfer?
A careless worker has cut each of the entire stock of 3 m lengths of mild steel rod randomly into three pieces.
The manager needs to ship some long pieces of mild steel rod.
What is the maximum length she can guarantee to her customers?
When Andy plays against Candy in their favourite game, the odds are 2 to 3 that he will win.
What is probability that Candy wins three games in a row?
This strange looking series was found in a Jolley collection from 1925. It references a book from 1899.
Such collections do not tell you how the series should be summed.
What is S?
S is a geometric series. The 3-dots (an ellipsis) means carry on in the same way without limit.
What is its value?
Hint: What do you get if you subtract S/3 from S?
We need a biased generator of random numbers: ⅓ probability of 0 and ⅔ probability of 1.
How many throws of a die are needed to be sure that the given probability condition happens?
In a special-needs class there are 5 invisible students who are required to arrange themselves on the 10 seats in the front row of the classroom, there being no other students in this particular class. Invisible students are to be considered as indistinguishable from each other.
In how many different ways can the students arrange themselves?
There are 10 discs of identical weight, colour, texture and so forth such that they are indistinguishable to your gloved hand. You randomly draw a number of these discs from a bag in such a way that you do not see the number written on any disc. The discs are uniquely numbered from 0 to 9.
How many discs must be drawn from the bag to guarantee that you can sort them and find a pair which sums to 9?
You intend to produce a code string of 5 letters, the order of the letters being important. The only letters available to you are { A, B, C, D, E, F, G, H }, each letter only being used once in any particular string. The first code letter has to be a vowel. The second, third, and fourth code letters are otherwise unrestricted. The fifth code letter has to be a vowel.
How many code strings can be produced?
Consider the set S = {x, y, z}
where x, y, and z are letters.
We wish to evaluate a new set, P, which is defined as the power set of S.
A power set is defined as the set of all subsets of the set in question.
Which is the correct expression of P?
HINT: Read the question slowly, carefully, and several times if necessary.
Estimate the number of small balls that would fit anywhere inside a grey ring having twice the internal diameter of the grey ring shown in the image.
A school bus is considered as "overloaded" if there are more than 50 students in the bus.
One hundred school buses, some of them overloaded, arrive at a school today.
What is greater?
How many ones are there in the binary representation of the given product of a Fermat number and a Mersenne number?
An official creates two small sets of playing cards in a special way. From one complete deck he randomly chooses 3 cards for one player, then randomly chooses 3 cards for the other player from another complete deck, but in such a way that no two chosen cards are the same. To be clear, the 7 of hearts is distinct from the 7 of diamonds. He then picks the same card from both decks, giving one to each of the players. Each player now has 4 cards which are shuffled in the usual way.
On command from the official, both players place a single one of their cards on the table, side by side with the other player's card, both face up. If the cards match exactly the players win. After all cards are played, if no match has been found, the official wins.
What is the chance that the official wins (without cheating)?
The grey circles are on a 10 foot grid and have a radius of 1 foot. The yellow disc has a radius of 3 feet.
In order to teach their children about probability theory, and PI, the teachers arrange to get the kids to throw the yellow disc thousands of times, blindfolded, and they count how many times a grey circle is touched or covered by the yellow disc. Multiple classes try this experiment and the results are combined.
The result is a probability value P.
How do you determine PI from P?
Gerry buys flowers and asks Jane for a rendezvous to tell her something very important.
They agree to meet after 8:00 p.m.
Gerry arrives at some time between 8:00 and 8:15, waits 5 minutes, and leaves if his love is not there. Jane does the same.
Estimate the probability that they meet and he asks the important question.
There are 16 students in a class.
The teacher randomly chooses a pair of students.
The probability that two girls have been chosen is 0.3.
What is the probability that a boy and a girl have been chosen?
Seven children sit on a bench. The first girl tells a number to the second girl and she tells another number to the next kid.
If the first girl starts with 7 and each next girl either adds seven to the number she received or subtracts one from it, what is the final result?
Gerry sent three letters to three girlfriends. He did it 3 times this week.
The postman put the letters into three different boxes without looking at the name of the recipient.
What is the probability that all girls received all their letters?
Five fair coins are tossed, all at once.
What is the probability that the result is either exactly 2 heads or exactly 3 heads?
Arty and Barty play a game of chance using three fair standard 6-sided dice. Arty pays $8 to play each game. $Barty pays $1.
For each game Arty states three numbers as possible values on any of the dice. Barty rolls the dice. If any of the stated numbers shows up Arty takes the pot, that is the $9 total sum. Otherwise Barty gets that money.
Over a long run of these games, with Arty always stating the numbers, and Barty always rolling the dice, who wins (gets more money)?
Amongst the possessions of the recently deceased pirate Long John Parrot was found a treasure map detailing the location of the island on which a treasure hoard has been buried. Sadly rats have chewed out the detail of the location on the island.
There are 50 palm trees on the island, and pirates always use palm trees as reference points. Long John Parrot always chose simple compass directions (North, South, East, and West) from any tree, and only used an odd number of (human) paces up to (and including) 7 as the distance.
How many holes need to be dug to give a 50:50 chance of finding the treasure?
A lady sent four letters to four gentlemen, but randomly placed the letters in the envelopes with their names.
Estimate the probability that only Sir Arthur receives his letter and nobody else.
On his multiple choice exam, where each question has 5 possibilities, and there are 100 questions, Sherlock scores zero. He has answered every question.
Estimate the probability that this happened randomly.
We wish to convert the word STOP into the word POST by swapping non-adjacent pairs of letters.
To be clear, we cannot swap the OP to PO as the letters are adjacent.
How many swaps are needed to perform the required action?
I flip three coins in a trial.
On average, how many trials do I make before three heads have turned up?
Ann hits a small target 80% of the time and Bob only 40% of the time.
After they shoot together, just one arrow each, only one of them hits the target.
What is the probability that it is Bob's arrow?
At an international conference there are 100 delegates, all of whom speak their own distinct languages.
How many translators are required so that each delegate can understand every other delegate without multiple levels of translators?
Ahmed and Betilda play a game using a single standard 6-sided die. Whoever throws a 6 first wins. Ahmed throws first.
What is the probability that Betilda wins?
Jane and Gerry take turns tossing a coin. The first one to toss tails wins.
If Jane starts first, what is her probability to win?
Strictly speaking we have asked what you get when you raise (4k + 1) to the power of N, where k is an element of the set of positive integers, and N is an element of the set of natural numbers (which are also positive integers, except maybe zero is also optionally included). But we wish to consider numbers of the form 4k + 1 such that in each bracket the value of k could be different. We are just too lazy to write this out as (4a + 1) x (4b + 1) x (4c + 1) x ... for a, b, c, ... all positive integers.
Given that again we are too lazy to use yet another variable name, and that we abuse the equals sign by using a different value of k on the right-hand side of the equals sign compared to the left hand side, what is the answer?
A teacher randomly splits six kids into two groups of three.
What is the chance for Alex to be in the same group with his secret love Betty?
It is certain that a snow storm either hits the Engadin region in Switzerland or not tomorrow.
Two independent antique weather stations predict the weather with equal probability ¾ and they both say that the storm will happen.
Based on the given information find the probability that the storm will happen in the region tomorrow.
Austin and Bart organize a duel.
They will take turns shooting at one another until one has been hit. We know that
1) Austin can hit Bart only 25% of the time,
2) Austin starts first, and
3) they have equal chances to be shot at the end of the duel.
What is Bart's probability of hitting his enemy in a single shot?
Regular tetrahedral (triangular-based pyramid) dice have four faces, numbered 1 to 4 inclusive.
Having thrown two such dice, the ratio max/min of the two values is chosen as the score.
How many of the possible permutations have scores which exceed unity?
A tetrahedron (triangular-based pyramid) has four faces. Dice can be made from regular tetrahedra, with the faces numbered 1, 2, 3, and 4.
Having thrown two such dice, the minimum value from one of them is chosen as the score.
What is the average (mean) value of the score?
Now H is some indeterminate but unreasonably HUGE value, and you are required to evaluate this expression symbolically.
If it helps, you could imagine that H is an integer with say 300 decimal digits.
What is the correct answer?
Hint: If the notation is unfamiliar to you, try the easier question first.
by Leslie Green
There are 40 ordinary coins on a table, with exactly 10 showing Heads.
You are blindfolded and wearing heavy gloves, so you can only feel the presence of a coin, but do not know which way up it is.
The coins showing Heads are randomly distributed across the table, and you do not know where they are.
By splitting the coins into two groups, and then turning over the coins in one of the groups, you are required to make the number of heads showing in each group equal.
What is the size of the larger group of coins?
We wish to form as many distinct (unique) sequences of the letters A, B, B, and A as possible.
How many are there?
Gerry tosses a coin and Jane tosses two coins.
What is the probability that Gerry gets the same number of heads that Jane gets?
Bernie has one fair coin which he will flip.
Ernie has ten fair coins which he will flip.
What is the probability that Bernie flips more heads than all of Ernie's heads combined?
Laura plays a game of dice with her younger brother Harry. Each has two fair 6-sided dice.
Laura's score is the lowest value of her two dice. For example, if she throws 3 and 6, her score is 3.
Harry's score is the highest value of his two dice. For example, if he throws 2 and 5, his score is 5.
P(L) is defined as the probability that Laura's score exceeds Harry's.
P(H) is defined as the probability that Harry's score exceeds Laura's.
What is the most that we know?
Your temporary (supply) mathematics teacher hates children, and just wants the class to be quiet. He therefore sets the class the task of throwing 9 ordinary fair dice, all at once, counting the number of spots on each upward face, summing the numbers, and recording if the sum is odd or even. The class is expected to perform this experiment 100 or more times in order to establish, with confidence, the probability of the result being odd.
You, being equally bored by this time-wasting experiment, decide to evaluate the probability for yourself.
A group of four friends decide to exchange gifts. Each person writes his/her name on a piece of paper and puts it in a hat and then each person randomly draws a name from the hat to determine to whom his/her present goes.
What is the probability that at least one person draws his/her own name?
Four pool balls, numbered 1, 2, 3 , and 4, are placed in a line from left to right in such a way that not more than one adjacent pair are in increasing order from left to right.
The sequence: 4, 1, 2, 3 is not allowed because there are two adjacent pairs in increasing order ([1 - 2] and [2 - 3]).
How many of these sequences exist?
The ACME Budget Ballot Sorting Machine is the cheapest machine on the market, and therefore widely used by different States. It only has two possible output bins. Any ballot paper entering the machine should end up in one or other of these two bins, but a well known defect allows ballot papers to slip into the gear train and get chewed up into dust.
If five ballot papers are fed into the machine, how many different sets of bin counts are possible?
James shares a secret with two boys in his school.
Five minutes later, each of them shares the secret with two boys who have not known the secret yet. Five minutes later, these 4 boys do the same and so on.
If the secret spreads with the same speed, estimate how many boys know the secret in 45 minutes.
There are four bags of rope, containing 10 m, 20 m, 30 m, and 40 m respectively. Due to a misunderstanding in the packing department, they all have the same weight and shape, and all are labelled only "ROPE". The packages are effectively indistinguishable.
Having taken two bags at random, what is the probability that you have exactly 50 m of rope?
Find the sum of odd numbers up to some value n, where n is also odd.
Feel free to spend as long as you like working out the answer for yourself, rather than just picking a workable formula.
We write the probability of event A happening as P(A). Let's suppose P(A) = a, where a represents a numeric probability value.
Suppose we have three possible independent events A, B, and C, and we wish to know what the probability of any one, or any combination, of them happening is.
Write P(A) = a, P(B) = b, P(C) = c.
Further suppose that at least two of a, b, and c are small, by which we mean less than 0.01 for example.
How do we combine a, b, and c to approximate the required probability?
We consider mission-failure of our new rocket based on only two critical systems: the fuel pump and the fuel controller. Fuel pump failure has been assessed as having a probability of 1/100. Fuel controller failure has been assessed as having a probability of 1/25. Both sub-systems have to work correctly to prevent mission failure.
Based on these statistics alone, what is the probability of mission failure?
I roll three dice many times.
Among the four results below what is the more frequent sum of the three numbers?
A dart is thrown at a dartboard in such a random manner that the probability of hitting any particular point is the same as the probability of hitting any other point. If the dart doesn't hit the board we throw again and again until the board is hit. The probability of hitting the board is therefore 1.
However, there are infinitely many points on the board, and the probability of hitting any particular point (with infinitely sharp darts) is zero.
What does this mean?
A perfect square is an integer formed as the square of another integer.
What percentage of all the counting numbers are perfect squares?
The 6 playing cards shown are jumbled up and placed face-down.
A player, who did not see the cards being placed face down, who uses no trickery or deception, and who gets no clues from anyone, takes a first card. Without looking at the card, he throws it away, never knowing what it was.
What is the probability that the next card is red?
You have learned statistics at school in order to pass an exam.
Are they of any practical use once you leave school, or is it a subject you can quickly forget?
The six counters labelled 1-6 are randomly placed in a line which we can consider as a string of digits. This can be done in exactly 6! = 720 different ways.
How many of these 720 permutations contain a sub-string of at least four digits, such that in this sub-string each digit is one greater than the immediately preceding digit, looking from left to right?
The five counters, marked '1', '2', '3', '4' and '5', are randomly place in a line.
What is the probability that they form a string of at least 3 digits, each increasing by 1 from left to right?
You have counters labelled 1 to 3 as shown, and three identical blank yellow counters.
Given that the labelled counters always have to be placed in increasing order to the right (as shown), how many different arrangements are possible?
You start on the green square at the left. You move to the red square at the right. Each move is either up, down, left, or right. You are not allowed to step on the black square.
If it were not for the black square, there would have been 120 paths, all of the same minimal length.
When the black square is considered, how many minimal length paths are there?
A five-headed Hydra has appeared on the Japanese mainland and is wreaking havoc, killing all the villagers in its path. A battalion of brave katana-wielding samurai has been dispatched to defeat it.
Their strategy is clear. On the third beat of the battle drum the nearest samurai must simultaneously cut off a Hydra head, one samurai per head. If all heads are severed the beast dies. If even one head remains attached, however tenuously, all the other heads grow back within ten seconds – except each cut head is replaced by two new heads.
These samurai are highly skilled swordsmen, but hydra necks are not stationary, waiting to be cut. Therefore the probability of cutting an individual head off is only 50%.
What is the probability that the beast is vanquished on the first round of head-chopping?
There are 5 counters in a bag.
Three are Argentinean (blue) and two are Brazilian (green).
Three counters are randomly picked out of the bag, one by one.
They are not returned to the bag.
What is the probability that all three counters are Argentinean?
Five unique positive integers have an arithmetic mean of 10. Two of the numbers are given as 5 and 30.
How many variants of the other numbers are there, given that order does not matter?
Three boys each have a tendency to tell the truth ¾ of the time.
One flips a fair coin that they all see.
They all say it's a Head.
What's the probability it is a Head?
John tosses one fair coin, and Mary tosses two.
What is the probability that Mary gets less heads than John?
There was a tradition in old Russia. A would-be bride gathers six long pieces of straw and grasp them in her hand. She then randomly ties pairs of knots on the top and the bottom. Since there are six blades of grass sticking out above and below the hand, she will tie three knots on the top and three knots on the bottom. If she forms one big ring, she gets married soon.
Estimate the probability that the girl will get married soon.
The problem is mentioned in The New York Times NUMBERPLAY and it is credited to Sunil Singh.
John threw 2 fair coins and Mary 4 coins.
What is the probability that Mary has more heads than John?
Four gentlemen visited a windy street wearing identical hats when a gust of wind blew off all the hats. Each of them picked up a hat and placed it on his head.
What is the probability that exactly two of them have their original hats?
Jane expressed the value of π = 3.14159... (decimal) using the base 2 (binary) system:
11.0010010000111111...
Which symbol does the notation have more of?
A thumb war tournament had four players: Amanda, Bob, Christine, and David. They form two teams of two players.
If the players are assigned so that all possible arrangements are equally likely, which of the following arrangements is more likely?
At an annual international conference there is one delegate from each of the 28 member nations. It has been decided that no delegate shall sit next to a delegate they have previously sat next to around the huge circular table. This scheme is designed to avoid bias.
For how many years in total can this rule be applied?
(Note that it is the country which is relevant, not the individual delegate, so new delegates can be substituted without changing the rule.)
In the ancient encryption scheme shown, a letter can be represented by a number. This alone is not very secure. However, if that number is formed from three other numbers, which are in turn represented by letters, things get more difficult for the "code-breaker" (cryptanalyst).
Consider the letter "E". It has a value of 5. We can form the 5 from 6 + 2 - 3 = 5. We can therefore represent the plain-text "E" by the cipher-text "FBC". The pattern used is fixed for each particular use, in this case two numbers are added, and the third is subtracted. The cryptanalyst now has to find both the cryptographic table values and the mathematical pattern used.
Crucially, a letter is not always represented by the same cipher text, defeating the cryptanalytic technique of "frequency analysis", which uses the fact that certain letters appear more frequently in a given language than others.
How many ways are there to encrypt "E" using the above rules?
The 6 x 6 grid shown consists of 36 little squares, and a blue border which we will neglect. Having drawn a single square, we wish to create the full grid by a series of copy and paste operations. Each copy is one operation, and each paste is one operation. It is permitted to do one copy, and multiple pastes of that copied pattern, in order to reduce the operation count.
What is the minimum operation count required to create the whole grid, given that squares cannot be deleted once pasted?
You are going on a hike and wish to pack as much stuff as possible into your rucksack, subject only to a weight limit of 60 pounds -- which must not be exceeded. All weights are measured in "pounds", and the word will not be mentioned again. You have 7 items that you could pack, and they have weights of 4, 25, 9, 35, 5, 7, and 30.
How many different combinations of stuff can you pack, subject to the conditions mentioned above?
In an ancient game (which we have just invented) you throw two dice, one red and one green. The value on the red die is used to display the number of gold doubloons that you win. If and only if the number on the green die is equal to the number on the red die, the displayed score is doubled.
What is the average win per roll of the dice?
I can climb three steps in exactly four different ways.
How many different ways can I climb N steps?
We have 3 shapes to be placed along a line.
The order is which they are placed is important.
Each shape can only occur once along the line.
Each shape can be one of three colors: blue, green, and purple.
The hexagon is always green.
Each color only occurs once along the line.
How many different arrangements are there?
Gerry placed 8, 5, 4, 2, and 1 golden ducats in five small bags, so that nobody can detect their content.
What is the probability that there are more than 10 ducats in two randomly chosen bags?
A random variable can take any value between 1 and 2, inclusive, with uniform probability.
What is the probability of a value of 1.49?
by Leslie Green
Imagine that the numbers in the image are written onto a cylinder so that 1 is next to 2, but also next to 4 -- going the other way.
A swap consists of interchanging the contents of two adjacent boxes. For example 5 and 6 are adjacent, and they could be swapped in one operation.
We want to sort this set of numbers into increasing order, but they are allowed to increase left-to-right or right-to-left. We always check for increasing order starting from 1, but 1 does not have to be in the leftmost position shown.
How many swaps are necessary?
(At this level, correctly guessing the number of swaps without noting which pairs are swapped is not to be considered as a correct answer.)
by Leslie Green
Imagine that the numbers in the image are written onto a cylinder so that 1 is next to 3, but also next to 4 -- going the other way.
A swap consists of interchanging the contents of two adjacent boxes. For example 5 and 2 are adjacent, and they could be swapped in one operation.
We want to sort this set of numbers into increasing order, but they are allowed to increase left-to-right or right-to-left. We always check for increasing order starting from 1, but 1 does not have to be in the leftmost position shown.
We know from an earlier question that only 4 swaps are required to reverse the rotational direction of an ordered set of five people (or numbers).
How many swaps do we need to perform in the present case?
by Leslie Green
Five friends randomly sit down at a round table. Ann notices that to her right are Ben, then Carol, then Derek, then Emma.
Ben is appalled to find out that these four friends are sitting alphabetically, but in an anti-clockwise direction as seen from above. Being scientifically minded they all set about working out how many interchanges of adjacent people are necessary to correct their unnatural rotational orientation.
In simple terms, only people sitting right next to each other can swap places, and obviously they need to orient themselves alphabetically, in a clockwise direction, as seen from above.
How many swaps are needed to correct this distressing situation?
by Leslie Green
On an island, every 16th mathematician is a philosopher.
Every 4th philosopher is a mathematician.
Every 2nd philosopher is a writer.
Every 4th writer is a philosopher.
Every 6th writer is a mathematician.
Every 12th mathematician is a writer.
Can you find the relation of the numbers of mathematicians (M), philosophers (P), and writers (W)?
We have balls which are indistinguishable from each other, apart from their color. On command we put a single ball into a standard opaque bag, designed specifically for mathematical problems. With probability p the ball is red; otherwise it is blue. We perform this operation twice, meaning there are now two balls in the bag.
Some stooge is then employed to randomly withdraw a ball from the bag. The stooge has no name or gender as that would cost more.
What is the probability that the withdrawn ball is red?
by Leslie Green
With equal probability we randomly pick a red, white, blue, or green counter from a bag.
If we picked blue then we throw it away in disgust, and randomly pick again.
If we picked red or white we are done.
If we picked green then one third of the time we choose red, and two thirds of the time we choose white.
What is the probability that we end up with red?
by Leslie Green
A committee is in the unusual position of having to make a random decision between 3 fairly equally supported proposals. To make the decision they each write a number within the given range on a piece of paper.
The sum of the written values is taken modulus 3 to give three distinct results.
Which range cannot be used?
For no clearly defined reason, two balls have randomly been placed in a bag. The balls were randomly chosen from red, white, and blue options, again for no clearly defined reason. The possible combinations in the bag are exactly six: RR, RW, RB, WW, WB, BB.
Which combination is most likely?
by Leslie Green
On average, how many times do you need to throw a fair die to get a 6?
(In other words, how many times do you need to throw a fair die to give a 50% chance of getting a 6?)
If a chord is selected at random on a circle what is the probability that its length exceeds the radius of the circle?
Assume that the end points of the chord are uniformly distributed over the circumference of the circle.
An ancient clay tablet has been found with strange markings on it. Sadly the bottom part has broken off.
There are 4 broken pieces available.
Which one belongs where the red question marks are shown?
We have a strange cat who will only eat 90% of the food given to her. Each time we feed her, we give her 10% less, in the hope that she will not waste 10%. But no, she only eats 90% each time. We can only suppose she used to be a Buddhist mathematics professor.
Just for fun we wonder how much food she would have consumed if this process were to continue indefinitely.
(We label the initial offering of food as 1 unit.)
You have a normal fair six-sided die (see the picture). You have a normal fair two-sided coin.
You throw the die and toss the coin, applying a mathematical operation on the value from the die to obtain a result.
It is required that the result is a number between 1 and 12, and the probability of any of these 12 possible results is equally likely.
What is the mathematical operation?
In a particular country 80% of the population speaks German, 20% speaks French, and 50% of the population speaks the foreign language - English.
Two people talk to each other during a national congress.
If the linguistic proportion is fair for the congress, what is the probability that the two people understand each other?
How many triangles are formed by five intersecting lines?
There are no points where more than two lines intersect and no two lines are parallel.
Try to figure out a procedure to calculate the answer.
Special string is available which can be cut up by a secret process into as many equal pieces as you would like. Three pieces of such string are available in 10 mm lengths.
Gerry gets string A after it has been cut in half 30 times, each time discarding one of the pieces.
Jane gets string B after it has had the last two thirds cut off, 19 times.
Tom gets string C after it has been cut into H equal pieces, and all but sqrt(H) of these pieces have been discarded, before the remaining pieces are put back together by a secret process. (Had all H pieces been available, the string would have been restored to its original length.) H is not precisely defined, but just suppose that it is adequately HUGE, say at least as great as 1E40.
Who gets the least string?
by Leslie Green
Estimate the number of triangles (with non-zero area) with each of the three vertices at one of the dots in the diagram.
The diagram shows a circle that has been divided into five sectors of different sizes. The sectors are to be painted red, yellow, or green. Any two sectors which share an edge are to be painted in different colors.
In how many ways can the circle be painted?
Evaluate the infinite series, S.
(In this context, the three dots, an ellipsis, means continue in the same pattern without limit.)
There are 7 soldiers in a squad. Every evening three of them are on guard duty, but no two of them are on duty at the same time more than once.
For how many evenings can this pattern continue?
There were 3 branches on an apple tree in the first year. At the beginning of every following year the owner cuts one third of the old branches and 2 new branches grow from each branch that is left. He always cuts branches that have no other branches.
Estimate the number of branches in the eleventh year.
Marek, the pan-dimensional super being, is bored. She therefore decides to play a game. She moves from her starting point in a sequence of instantaneous jumps, all in the same direction. These jumps naturally violate many of the known "Laws" of the infestation known as humans.
She starts at 1 minute to midnight. 1/2 minute later she jumps 1/2 a light-year. 1/3 minute later she jumps 1/3 of a light year. 1/4 minute later she jumps 1/4 of a light year, and so on.
How far from the starting point is she at midnight?
(NOTE: Do not consider relativistic temporal distortion.)
by Leslie Green
You have 10 gold coins, exactly one of which is fake, and therefore significantly lighter than the rest. You have a pair of balance scales which will balance for any equal number of true gold coins on both sides.
What is the greatest probability that you can correctly identify the fake coin in at most one balance operation?
by Leslie Green
In some distant future, the Paradise Hotel has H ordinary rooms available, each at the cost of sqrt(H) dollars per night. The madcap genius hotel owner, Dr Hilbert, has devised a brilliant scheme to increase the hotel's capacity by utilising orthogonal imaginary space. For each of the ordinary rooms he has created H-1 rooms with positive-integer imaginary-space existence: For example (1,0) is an ordinary room, but (1, i1) is the first in a long line of complex-space rooms. Each non-ordinary room costs $1 per night.
Due to expansion, H doubles every year. H was already HUGE!
What is the effect on the average room cost?
by Leslie Green
The expression shown in the image is probably unfamiliar to you.
In English you could interpret this as, "Just keep increasing n for as long as you like until the expression in square brackets settles down to some sort of definite value."
What is that limiting value?
by Leslie Green
Jane is on holiday in a strange and far away land. She can't quite understand where she is because the locals call it England, Great Britain (GB), and the United Kingdom (UK) in a seemingly random fashion!
Jane has at least 20 of each of the local coins, valued at 1p, 2p, 5p, and 20p respectively. She is waiting to pay for a snack at a local shop, and wishes to pay with the exact change to show her mastery of the local currency. The snack costs 14p.
How many different ways can she pay with the coins in her pocket, given that all coins of the same value are considered identical, and she would just pass over the money as a handful of coins?
A coin collector has 7 large gold coins, 5 large silver coins, and 8 large bronze coins. His assistant has been told to remove them from the locked security cabinet and dust them, before returning them to their drawers. There are exactly 10 drawers, and the assistant successfully puts two coins in each drawer, albeit in some haphazard way.
If the collector opens a drawer and finds a gold coin, what is the greatest* probability that the other coin is also a gold coin?
*Consider the case when the assistant places the coins in a way that is favorable for the selection.
by Leslie Green
The arrow symbol is not quite an equals sign. It should be interpreted as:
(a) tends to
(b) becomes
(c) could be written as
(d) simplifies to
H is an indeterminately HUGE value. Think of any HUGE value you like, then multiply it by 1E300, and H will be a bit larger!
Are these relationships reasonable?
by Leslie Green
I have a very poor memory for numbers, so I like to hide my 4 digit PIN number in a 4 x 4 grid (in a fairly secure location), as shown. Each letter represents a digit in the PIN number. Each x is a random digit. All digits are black, and are the same size.
What I do remember is a little picture of the sequence in which the actual 4 PIN digits were positioned. Maybe clockwise in corners, maybe diagonal. The image shows just one such pattern, a sort of rotated 'L'.
How many such patterns are possible?
(There is no need for the digits to be adjacent).
We start at the top of the image on the purple square. At each step down the page there is an equal probability of going left or right, shown by the arrows. The blue squares are the end points. We would like to know the probability of going one step to the right before going two steps to the left. Notice that there is another purple square because if we go one step left, then one step right, we are back to where we started.
What is the probability of going one step to the right before going two steps to the left?
by Leslie Green
The geeky twins go for a walk in the woods. They reach a junction and have to decide which way to go. Having geeky parents, they both came armed with the digital random decision makers which they received for their birthday. Much better than tossing a coin or throwing a pair of dice, the Decidatron (by Pat Pending Ltd) can be dialled up to any arbitrary probability for a YES outcome.
Mary asserts that she will set her Decidatron to 0.40, and that she will go first. What this means is that she has a 0.40 probability of getting a YES every time she presses the button. They will take turns in pressing the buttons on their respective Decidatrons. They agree that whoever gets a YES first on their Decidatron gets to choose the path they will take.
What value does Carey have to set in order that both twins have an equal opportunity to choose the path?
HINT: try an easier one first.
by Leslie Green
An anagram is a word (or set of letters) formed by rearranging the letters of a different word.
For example, the word binary can be rearranged into brainy.
How many anagrams does the word Mississippi have?
Use all eleven letters and count all possibile sets, even if they are not real words with a definite meaning.
The blue ring is the patrol area for the cat, C, represented by the green dot. The central red dot represents the position at which the mouse, M, will emerge from its safe hiding place. We consider the cat and the mouse as dimensionless points for simplicity.
The cat has a uniform probability of being anywhere within its (blue) patrol region when the mouse randomly appears. The red ring is the danger zone for the mouse. If the mouse appears within the confines of the red ring it is munched (eaten).
Given that the blue ring has a diameter of D and the red ring has a diameter of d, what is the relationship between D and d that gives a 25% survival probability for the mouse?
by Leslie Green
Data Analysis:
Consider the common saying shown,
which can also appear using either 'me' or 'us' in place of 'you',
and has even been included in at least one pop song?
by Leslie Green
Gerry is sure that he knows the answers to half of 44 multiple-choice math questions on the SAT. There are five answer choices for each multiple-choice question.
Calculate his expected score if he guesses randomly each of the other questions.
We have 2N+1 coins, (N > 7), exactly one of which is a fake coin with either two heads or two tails (and we don't know which). We cannot spot this fake coin by looking at just one face.
Having tossed all the coins onto a flat surface we wish to flip individual coins over until they are all showing the same face (either heads or tails).
Given that we are extremely unlucky, what is the minimum number of coin flips necessary to achieve the stated goal?
(HINT: Try an easier one first.)
by Leslie Green
Janet is going to her friend's birthday party. Only 4 people in total will be attending, and one of them is her dreamboat Kevin.
Three of the attendees will sit randomly around the party table. Janet will definitely sit next to Kevin if she can.
What is the probability that Janet can sit next to Kevin, given that she will be the third person to arrive?
by Leslie Green
You toss three coins, all at once. At least two of these coins are fair and unbiased. At most one of the three coins is a fake, a magician's coin, fixed with either two heads or two tails. You don't know if you have tossed such a coin, and if you have which type of coin it was.
In the worst case, what is the minimum number of coin turns you need to perform in order to get all the coins with the same face upwards (eg all heads).
A coin turn is defined as turning the coin over such that for a fair coin, a head would become a tail.
by Leslie Green
We have a circle with 4 points randomly distributed around its circumference. We join all pairs of points by line segments, without re-using any point. Each configuration therefore uses exactly 2 line segments.
For any particular distribution of points, how many unique configurations of line segments exist?
by Leslie Green
The average monthly income of Alex (A) and Betty (B) is $20,000.
The average monthly income of Betty and Craig (C) is $30,000 and
the average monthly income of Craig and Alex is $10,000.
What is the Betty's monthly income?
Six points are randomly chosen at the edge of a pizza.
Three straight cuts are made by passing through two of the points, never the same.
How many different ways are there to cut the pizza under such conditions?
A large piece of balsa wood is carved into 4 separate solid pieces. One piece is a solid sphere. One piece is iceberg shaped. One piece is shaped like a boat. One piece is shaped like a submarine.
One of the four pieces is picked at random and dropped into a bucket of water.
What is the probability that it floats?
by Leslie Green
Suppose that you meet 1000 people in your time at college. Further suppose that 40% of those are male. Further suppose that 20% of those are stylish. Further suppose that 10% of those have trendy hairstyles.
How many stylish men with trendy hairstyles do you meet at college?
by Leslie Green
Two fair dice are thrown. The smaller value is subtracted from the larger value.
What is the probability of getting a result of 1?
If Gerry flips a fair coin two times, the probability of obtaining 1 head and 1 tail (in any order) is 50%.
If he flips the coin four times, what is the probability of obtaining 2 heads and 2 tails (in any order)?
Bob is going out on an unusual plumbing job, and doesn't know what extra tools to take with him. He has 3 specialist tools which he could take, and all of them are quite large.
How many different combinations of these specialist tools can he choose from?
Susan wants to write a function 'digits()' as part of her computing project.
The intention is that N is a counting number from 1 to 100,000 (inclusive) in ordinary decimal format, and that the function digits(N) returns the number of digits in N.
The function log10(V) returns the base-10 logarithm of V.
The function ceiling(k) returns the next integer larger than or equal to k.
Comment on the function for a large number of random inputs in the given range.
by Leslie Green
Consider the summation S shown in the image with
a = 1/3
N = 10.
How many ones are there in the fractional ternary representation of S?
Ternary means "base 3".
by Leslie Green
The engineering formula book gives a sum between the limits of 0 and N-1, but you need the sum between the limits of 1 and N.
What is the value of T?
by Leslie Green
Fifty 25-year-old young women and fifty young men of the same age arrive on an uninhabited island.
After 100 years, what will be the ratio (final population) / (original population), given that every couple always has 4 children (2 boys and 2 girls) at the age of 30, and everybody lives 88 years?
The image shows the proof that the sum of the infinite series S has a value of -1.
What is your response?
by Leslie Green
In a democratic country, 75% of the electorate vote in a particular election. The candidate 'Rob the Slob' wins the leader's election with 60% of the vote.
What proportion of the population definitely wanted Rob the Slob as their leader?
by Leslie Green
Jane spends 20% of her income on taxes and 25% of the remainder on rent.
What percent of her income does she spend on rent?
This is a typical GMAT question.
If one of my tortoises crosses a road in ten minutes on average, estimate the probability that all of my ten tortoises cross the road in ten minutes or earlier.
There is an old saying,
"A bird in the hand is worth two in the bush".
In order for these two assets to have equal value, there must be some defined probability of capturing a bird from the bush.
What is this probability, assuming that each bird-capture is an independent event?
by Leslie Green
Alice, Caroline, Emily, Bethany, and Danielle together have 18 mathematics text books.
Danielle and Alice have 6 between them.
Emily, Danielle, Caroline, and Alice have 15 between them.
Danielle, Alice, and Emily have 8.
How many mathematics text books does Caroline have?
by Leslie Green
A tree grows quickly in its first year, then its annual growth is always half as much as in the preceding year. It reaches its maximum height at the age of 100 years.
How many years does it take for the tree to reach half its maximum height?
The figure shows a map of a part of a city. Every morning John walks from his home H to his school S every time randomly choosing either north or east if there is such a choice.
What is the probability that he goes through the intersection P where Mary's father works as a policeman?
Matilda is an inquisitive child. She takes a block with a mean height of 100 mm and sets it on the hard floor. She supposes that its height has a Gaussian distribution with a variance of 4.5 mm. She now sets another of the same type of block on top of the first block.
Being quite advanced for a 9 year old, she now wonders what the distribution of heights above the floor would be for lots of kids performing the same activity.
by Leslie Green
Once a year a dragon drinks some of the Elixir of Life from the Magic Lake.
The Elixir is never replaced, and never evaporates or leaks from the Magic Lake.
At the end of the first year the dragon drinks 1/2 of the Lake, which was full beforehand. At the end of the second year the dragon drinks 1/3 of what is left, and so on.
In which year does the Magic Lake run dry?
You are (truthfully) told that a bag contains one RED ball and two other balls which are some random (and unknown) combination of BLACK balls and YELLOW balls. Before picking a single ball from the bag you will of course be blindfolded. All balls have the same weight, size, composition, rotational inertia, and surface texture. To cut a long story short, your choice will be purely random.
You will be rewarded handsomely if you correctly choose the outcome of this task. Which is your best choice?
(We are using the ! sign to mean "NOT".)
by Leslie Green
The image shows a spinner, a random selection device with 6 edges upon which it can fall with equal probability.
If the spinner lands on 1, 2, or 3, that is Event A. If the spinner lands on 2, 3, 4, or 5, that is Event B.
But these Events conflict with each other, so we also say that in the event of a conflict we toss a fair coin and Heads means it is actually Event A.
In the case that neither Event A nor Event B occur, we re-spin until one of these Events does occur.
What is the probability of Event A?
by Leslie Green
We select a set out of the integers from 1 through 17 inclusive.
None of the numbers is double another number in the set.
What is the largest number of integers we could have in the set?
The TV studio staff writers are bored with their action hero star and wish to bump him off to boost the ratings. In an unusual set of mutually exclusive and exhaustive outcomes, they plan to have him fall out of a helicopter with a probability of 1/3, get eaten by a crocodile with probability 1/4, contract a particularly fast acting virus with probability 1/5, get run off the road in a car chase with probability 1/6, or get thrown off a very large building.
What is the probability of him getting thrown off a very large building?
by Leslie Green
Over the period in question, inflation is running at a constant rate of 5% per year.
House prices are rising at 5% per year above inflation.
How long does it take for the dollar value of a particular house to double, everything else being unchanged?
by Leslie Green
Five students randomly choose their places on a very large bench. Jane comes first and chooses a place. Any future person will only sit in a seat next to someone who is already sitting down.
What is the probability that Gerry and Jane sit together, given that Gerry arrives last?
In an experiment the ratio X is evaluated as 4.00 with an estimated uncertainty of ±10%.
The result is evaluated from the equation
R = 1 + X.
What is the uncertainty in R?
Twenty-six students are randomly assigned to two equal-size teams.
What is the probability that Mary and John are in different teams?
In this game you are going to pick two cards at random from a pre-selected group of cards. If they both have the same color (colour) you win.
Clearly you want to win!
Which set of cards gives you the best chance of winning?
(Note: if you don't know where to start on this problem, try an easier one first)
by Leslie Green
John randomly chooses a piece of paper from a collection of green, yellow, blue, violet, and rose sheets. There is an equal probability of choosing any particular color.
At exactly the same time, Jane randomly chooses a pen from a collection of green, yellow, blue, violet, and rose pens. There is an equal probability of choosing any particular color.
Neither John nor Jane know what the other has selected.
Jennifer takes the pen and paper and writes a note.
What is the probability that the writing will be visible, given that writing using a pen of the same color as the paper will not be visible?
by Leslie Green
Gerry throws two dice.
What is the probability that the product of two numbers is even (evenly divisible by 2)?
This is a single sequence where there is a mathematical rule to go from one position to the next. There are no alphabets involved, this is a purely numerical problem.
1, 8, 4, 0, 7, 3, 10, ...
How many more steps before the 10 is repeated?
by Leslie Green
The probability of a man hitting a target is 1/3.
If he throws 3 times, what is the probability that he hits the target at least once?
What is the value, written in base 7, of the sum of the two numbers written in base 3 and 6?
123 + 456 = ??7
What is the fewest number of people that could have visited the open-air movie show, if exactly 0.8% of the people did not watch the film until the end?
In the field of complex numbers we have a number which consists of a real part and an imaginary part.
The imaginary part is some multiple of i, where i is the square root of minus one.
Given Z = 4 + 3i
and W = -3 - 2i
What is the real part of the product of Z and W?
We have been told that the local population of men have an average height of 66 inches, and that their height has a standard deviation of 2 inches.
Estimate what proportion of these men have a height in excess of 63 inches.
by Leslie Green
In Timmy's school there are lots of girls of a similar age to him. He estimates that about half are natural blondes and half are brunettes. There don't seem to be any black haired or ginger haired girls for some reason. He has been told that there are roughly as many girls with blue eyes as there are with brown eyes, and other eye colours are also not present.
Estimate the probability that if he picks a girl from his school at random she will be a natural blonde with brown eyes.
by Leslie Green
The cipher-text shown would be very difficult to decipher by hand if it were not for one crucial mistake. The sender has addressed the recipient by name, and we have guessed that the message is for Betty.
What is the message about?
by Leslie Green
Each of the five keys fits exactly one of five locks.
Imagine the worst scenario: how many times do you test a key in a lock in order to match up all the keys and locks?
If you can deduce that a key will match a particular lock, there is no need to do another test to confirm it.
I keep drawing randomly 1-cent and 2-cent coins until I reach or exceed 1 Euro. The first coin is equally likely to be 1 or 2 cents.
What is the probability that the last coin is 1-cent?
You spin a wheel and it randomly lands on $1, $2, $3, or END. If you land on $1, $2, or $3 you get that money and spin the wheel again. You keep receiving money until you land on END.
How much do you expect to win on average?
Twenty children dance in a circle.
Everybody has a handkerchief, that he/she gives either to the left neighbor or to the right neighbor.
The choice is random.
What is the expected number of kids with at least one handkerchief?
Ten brave men are in the corners of a pathway that has the shape of a regular star.
At the same moment in time, each of them randomly chooses a direction and starts walking.
What is the probability that nobody meets another man?
If the number of women in a company was increased by 50%, and the number of men was decreased by 50%, how did the ratio of women to men change in the company?
Leslie Green asks:
I have an unbiased coin, a fair six-sided die, and an ordinary pack of 52 playing cards. I toss the coin, roll the die, and pick a card at random.
What is the chance that I don't get a head, roll a number less than 5, and don't get a heart?
6 numbers are required to have an average of 9999. The first five numbers are: 9999, 9998, 9997, 9996, and 9995.
What must the last number be to meet the requirement?
Hint:Look for a sneaky method, rather than doing long-addition or algebra.
Author: Leslie Green
The largest known prime number was discovered by a computer laboratory of the University of Central Missouri in 2016.
257,885,161 – 1
Estimate the number of digits in the number.
Note: log10(2) ≈ 0.3
Sammy the Squirrel hides acorns in groups of 3, 4, or 5 per stash, the actual number being pretty random. He creates 200 such stashes in the autumn. When winter comes he only manages to find 5% of his stashes.
On average, how many acorns does he find from his stashes?
Author: Leslie Green
Gerry aims at the center (centre) of the large target. He is sufficiently accurate that he always hits the target. He throws 2 darts and adds up the scores.
What is his chance to score 3 in total?
The tree pattern has four rows.
If the pattern continues, what is rightmost number in the tenth row?
Four nearly identical balls are numbered 1 to 4 and then placed in a bag. The bag is shaken so the balls are mixed up. Jane draws two of the balls from the bag. Jane wins if the sum of the numbers on the chosen balls is greater than the sum of the balls left in the bag.
What is the probability that Jane wins the game?
There are 110 coins in a pile, 77 heads up, 33 tails up.
I randomly choose 33 coins from the pile and put them in pile A, and I call the remaining coins pile B.
I flip all coins in pile A.
Where is the number of tails greater?
In a scene from a film, the boss (baddy) is sharing the ill gotten gains with his (comically stupid) henchman by dividing the gold pieces into two 'equal' piles.
"That’s one for you, and that’s one for me".
"That’s two for you (puts an additional one on the henchman’s pile), and one, two for me (puts an additional two on his own pile)".
"That’s three for you (adds one to the henchman’s pile), and one, two, three for me (puts three more on his own pile)".
The final round is "N for you, and N for me".
How many times more pieces of gold does the baddy get?
This unsophisticated theme has been repeated over decades, mostly for young children. An early version was with Bugs Bunny, 'Racketeer Rabbit' (1946).
There is a 0.0001% chance of winning the Jackpot in a lottery that immediately gives the result.
People lined up in front of a kiosk and nine people before Gerry in the line failed to get the Jackpot.
What's the chance of Gerry winning the Jackpot?
This year Samantha has become very fussy about her birthday present. A square box must be wrapped in blue paper. A round box must be wrapped in red paper. An irregular box must be wrapped in green paper.
The probability of her getting a round box is 50%. The probability of her getting a green box is 1/4.
What is the probability that this spoilt child receives a blue present?
Author: Leslie Green
A telephone is locked by a combination of five digits.
Each of the five combinations 15321, 18622, 79643, 73737, and 15597 contains exactly two digits in the right position.
What is the sum of the digits of the combination that unlocks the telephone?
Three online meteorology (weather forecasting) services have a tendency to make mistakes 1/3 of the time.
They all say that it will rain in the old Swiss town of Murten tomorrow.
What are the chances that it really rains tomorrow?
The St. Petersburg paradox was solved by Daniel Bernoulli in 1738.
A casino staff member tosses a coin repeatedly until it comes up heads. If heads appears on the first throw, the casino pays you $2. If it appears on the second throw, you receive $4; if on the third, you receive $8 and so on, doubling each time.
How much would the casino be willing to accept from you to consider the game favorable for it?
A country boy takes 2 identical ropes, mixes the ends, and asks his girl-friend to tie two pairs of knots on the top.
There is a superstitious tradition in their village that if the result is a ring, then their common life will be long and happy.
According to this tradition, what is the probability of them getting a long and happy life?
There is a simple repeating sequence of numbers:
1, 2, 7, 11, 12, 17, 21, 22, 27, 31, 32, 37, …
This pattern continues up to 1000.
How many numbers are there in the sequence?
Author: Leslie Green
The blue square is the plan view of an open-topped box with slippery walls. The washer, which is not shown to scale, is thrown into the box at random, and the washer always lands flat on the bottom. The length of the inside edges of the box is 12 inches. The washer’s outer diameter is 1 inch and its inner diameter is 7/16 of an inch.
What is the probability that the washer lands entirely within the bottom left-hand square (red)?
Author: Leslie Green
There are a few more than one thousand students in a school.
Given that exactly 48.00% of the students are girls, how many more boys than girls are there in the school?
John proudly works for a Secret Service earning $5,000 per week. By mistake his wage was reduced by 10% last week. To fix the mistake, his reduced wage is increased by 11% this week.
Who wins from the changes?
The Funny Pins bowling club consists of ten married couples.
The club has decided to create a mixed (man-woman) team for the next tournament by randomly selecting partners. They don't want spouses to play in the team.
How many different teams are possible?
A hospital runs a queuing system for non-urgent surgical procedures. This queue always has 100 patients in it and one patient is operated on each day (7 days a week). If a patient is not available on the day of surgery they lose their place to a brand new patient who would otherwise have gone elsewhere. (This is easier than rescheduling 100 appointments.)
Roughly 5% of patients are not available or do not show up for their procedures.
What is the average waiting time reported to the managers of the hospital?
(If a patient does not receive surgery their waiting times are not included in any statistics.)
Author: Leslie Green
In factories where food items are packaged, one clever technique for optimally filling bags is to fill 12 nominally equal hoppers with the food, then computer select the 4 which give the closest fit to the required weight. This is better than taking food items such as crisps and putting tiny broken pieces in to make up the required weight. It is also cheaper for the manufacturer to not greatly exceed the minimum weight, and effectively get paid less for each gram of food as a result.
How many combinations does the computer have to check to get the optimal selection?
Author: Leslie Green
You have four different coins about the same size.
You have only a balance to compare the weights of two coins.
What is the smallest number of weighing steps needed to sort the coins from the lightest coin to the heaviest coin?
The maximum capacity of a standard school bus is 72 passengers.
Estimate the weight of people in the bus if a bunch of fathers take the bus to visit their children.
Wikipedia says that the average weight of a male American is 195 pounds or 88.5 kg.
1 tonne = 1,000 kg
Kitty has 2017 marbles. The marbles are numbered from 1 to 2017. Marbles with equal digit sums have the same color.
How many different colors of marbles does Kitty have?
There are four sisters. If we leave out any one, the average age of the remaining three sisters will be 6, 8, 9 or 10.
What is the average age of all four sisters?
Split the number below into 3 parts such that their sum is at a minimum.
(In other words, make 3 groups of the digits without changing their ordering.)
What is last digit of the sum?
8912673450
The average of 9 numbers is 9. If a number is added to the set, then the new average becomes 10.
What is the value of the number?
Five students randomly choose their places on a bench.
What are the chances that Gerry and Jane sit together?
Kitty has one million marbles, which are numbered from 1 to 1,000,000. Marbles with equal digit sums have the same color and marbles with different sums have different colors.
How many different colors of marbles does Kitty have?
Jasmine has just been learning about the binary number system at school. On her way home she wondered how far she could count using just the four fingers on one hand, if a curled finger represented a binary 0 and an outstretched finger represented a binary 1.
To be clear, she was thinking about counting up from zero in whole numbers. How far could she get?
Author: Leslie Green
Gerry's rich uncle gave him $2 on his first birthday. On each birthday after that he doubled his previous gift.
What was the total amount that his uncle had given him after the tenth birthday?
The US frequently used banknotes are $1, $5, $20, $50, and $100.
Gerry has 4 of each of these banknotes in his wallet.
He enjoyed the dinner with his girl-friend Jane and decided to pay the $200 total cost himself.
In how many different ways can he pay $200 using his banknotes?
A bag contains six green, five gray, and four violet disks.
How many green disks must be added to the bag so that 75 percent of the disks are green?
Each time a bar of soap is used, its volume decreases by 10%.
How many times would I wash my hands with the piece of soap so that less than one-half its volume remains?
When three fruits are randomly taken from the box, what is the probability that only apples are left in the box?
One hundred workers complete a project in 100 working days.
Adding any positive number of workers proportionally decreases the team efficiency.
Which number of workers allows a similar project to be completed most quickly?
What is the probability of getting 5 correct answers on A+Click questions in a row if you have no ideas about the correct answers?
Lottery 6-digit numbers are from 000001 to 999999. A ticket is considered to have a “special” number if the sum of the first 3 digits is equal to the sum of the last 3 digits.
Find the sum of digits of the largest possible special number.
No digit is used twice in the six digit number.
Jane has three parrots.
She said that at least one is a boy.
What is the probability that there is a girl in the trio of parrots?
Despite advice to the contrary from his friends and parents, Timmy has decided on a new strategy to select future girlfriends. He has two “must-see” programs on 5 days of each week. He requires that any future girlfriend must match-up with at least 90% of these programs. Given that there are 20 TV channels available in his area, what is the probability of a match?
Author: Leslie Green
An escaped criminal has stolen a spaceship, and has just instantaneously jumped 1 light year away. On each successive jump she will only be able to jump half the distance of the immediately preceding jump due to heat build-up. Jump engines always take 1 hour to recharge.
My ship can only jump 1/2 light year, but it has a better cooling system so the jump distances drop-off more slowly. My maximum jump distances follow the sequence: 1/2 light year, 1/3 light year, 1/4 light year, and so on.
I can find her with my subspace-tracker. If she is closer than my maximum jump distance I can get close enough to remotely disable her jump-drive and capture her. My jump engines are fully charged.
Can I catch her?
Author: Leslie Green
Jake, being bored on a rainy Sunday afternoon, throws a pair of dice 500 times and keeps a record of the results.
What is the ratio of probabilities between throwing one five and all the rest twos, compared to throwing all threes.
Author: Leslie Green
Anton, the highest IQ house ant on the planet, is taking his regular nocturnal walk from one corner of the chess board (1,1) to the diagonally opposite corner (8,8), moving only right or up the board at each successive square. He remembers that on square (5,3) there is a tasty sticky residue, so any valid route must involve this square.
From how many different paths can he choose?
[HINT: You could try an easier problem first.]
by Leslie Green
Jane has a 5-liter bucket with 5%-fat milk and 3-liter bucket with 3%-fat milk.
What is the fat concentration in the milk if she pours both into a large bucket?
Leslie Green asks: "In a large village community there are regularly women giving birth to triplets, but surprisingly never twins.
The average number of babies per mother is 1.9
The local nurse visits the next recent mother on her list.
Which is the most probable number of babies she will find?"
Cletus is absolutely, definitely, the worst student the driving school has ever seen. When asked to drive at a steady speed he constantly accelerates too hard, overshoots the target speed, and then brakes too heavily. He has what the instructors call a "heavy foot".
By some miracle he manages to always hit the same top speed of 20 mph before braking in a continuing non-repeating pattern like the one shown.
What is his average speed?
Author: Leslie Green
Jane and Gerry compete in a best-of-three match.
If Gerry plays so that his girl-friend has a 60% chance of winning any particular game, what is the likelihood that she will win the match?
Martin has ten thousand dollars in a bank account. The interest rate for his account is 10% every year.
How much money will Martin have in his bank account after three years?
Gerry tosses a coin and Jane tosses 2 coins.
What is the probability that Jane has more heads than Gerry does?
The pie chart shows the number of men, women and kids in a county.
What is the ratio of children to women?
Buy 2 for the price of 3 and get 50% off!
What is the final discount you get?
'Buy, buy, says the sign in the shop window; Why, why, says the junk in the yard.' - Paul McCartney
I roll two dice, one with the left hand and one with the right.
If the left hand die gives an odd number, the overall score is zero.
If the right hand die gives an even number, I roll it again and again until it is odd.
The score is the sum of the two numbers, except for the previously mentioned case.
There are exactly 6 possible scores: 0, 3, 5, 7, 9, and 11.
What is the probability of a score of 3?
The problem was suggested by Leslie Green
If I put these balls in a bag and take two balls without looking, what is the probability that the two balls are white?
Gerry and Jane like to look at registration plates of cars that pass by.
They add the three digits and bet on a result.
Among the values below, what is the best sum of three digits to bet on?
You are rolling two dice and adding the numbers on the top faces.
What sum is better to bet on if you have no other choices?
The cost of living increases uniformly by 44% in a period of two years.
What is the annual percent increase?
The masses of sugar and water in my 200 g tea cup are in the ratio 1 : 9.
If I drop two 25 g cubes of sugar into the cup, what will be the new percentage of sugar?
(We assume that the cup does not overflow in the process!)
If you toss a dime five times and it lands heads up three times, what are the chances it will land heads up if you toss it again?
There are 800 students in a school.
The students' average age is 12 years and 60% of them are girls.
What is the difference between the sums of girls and boys ages?
What is the difference between 11 in decimal (normal) and binary systems?
A binary number is made up of only 0s and 1s.
Gerry estimates that two angles of a triangles have a measure between 50 and 70 degrees each.
If this is true, what would be estimate for the third angle?
A row containing just 9 keyboard letters on Gerry's laptop has just failed.
If his password consists of 2 letters, 4 numbers, and 1 special symbol, what is the probability that he will be able to log in?
John's family tree includes 10 generations.
What is the minimum number of people on his family tree if there were no intermarriages?
How many 9s are there in the result of the multiplication:
12345 x 99999 ?
Don't use the calculator.
Bob used to weigh 100kg.
His new stressful job causes him to lose 10% every month.
How much does he weigh after 3 months of work?
Javier Sotomayor (Cuba) is the men's record holder with a jump of 2.45 m set in 1993, the longest standing record in the history of the men's high jump.
Javier is 1.95 m tall.
What percentage of his height did he jump?
Data source : Wikipedia
If the first President of the USA George Washington took office in 1789 and the fortieth President Roland Reagan completed his duty in 2004 what is average duration of the term for the American Presidents?
Anna spent 120 Swiss francs in a Swiss souvenir shop. She was charged an extra 2% for paying with her credit card.
How much did she spend in USD?
Use an exchange rate of $1 = 0.96 Swiss Franc
The average IQ of 100 students is 101.
If the first ninety-nine students each have an IQ of 100, what is the IQ of the last student?
A school offers the following dinner choices:
First course: soup or small salad;
Main course: vegetarian, chicken, pizza, cheeseburger or fish;
Dessert: apple cake, doughnut, fruit salad, ice cream or milkshake.
How many school dinners will I have eaten before I have to choose a combination that I have already eaten?
One bag contains two white marbles, another bag contains two black marbles, and a third bag contains one white marble and one black marble.
I pick a random bag and take out a black marble.
What is the probability that the remaining marble from the same bag is also black?
After a 20% discount and 8% tax, a set of tin solders costs $540.
What was the original price of the set?
After a test of a code, John detected 2 errors and Mary discovered 3 errors in the same code.
There is one error in common.
Estimate the number of errors that are still undetected.
In the United States, the average life span is 65 years and the average female gives birth to 2.6 children.
The average age when a female bears children is less than 30.
How many descendants might an average female have at the end of her life?
Alex must match three different numbers chosen from the integers 1 to 33 in any order to win a lottery.
He bought 333 tickets, each with a unique combination of numbers.
What is the probability of winning?
Guesstimation
A man can jump up to four times the length of his own body.
Frogs can jump up to fifty times their length.
A flea can jump 350 times the length of its own body.
What is the average length of these three jumps?
I ask you to pick a number from 1 to 100.
I then ask you, “Is the number greater than X?”
You must truly answer yes or no until I tell you the number.
What is the maximum number of questions I ask you if I choose the optimal strategy?
Here X is the number of my choice.
Alex(A), Beatrice (B), and Craig (C) work on a project.
(1) Together, A and B can complete it in 10 days.
(2) Together, B and C can complete it in 12 days.
(3) Together, C and A can complete it in 15 days.
If Alex does it alone, how many days will it take?
If the probability of an accident occurring in 4 years is 0.9999, what is the probability of an accident occurring in 1 year?
If 2 salesmen can sell three luxury cars in 4 days, how many luxury cars can five salesmen sell to six clients in 7 days?
The binary numeral system represents numeric values using two symbols: 0 and 1.
How is the number 999 written in binary code?
An average baby is about 20 inches (51 cm) long and weighs approximately 6 to 9 pounds (2700 to 4000 grams).
How many times heavier than a baby is an adult?
John sent an email to his three partners.
Everybody answered and copied the answer to all others.
A total of 12 emails were sent.
How many emails are sent if there are 20 people communicating in the same manner?
Abbey has a dog called Abby.
She places 5 tiles with the letters of her surname into a bag.
She picks out one tile after another without looking.
What word has the greatest probability of appearing in the correct order from the beginning?
John is sick 12 days per year.
The probability of being sick on a Saturday or a Sunday is two times less than on any other day.
What is the probability that John will be sick on a Monday?
Jane and Gerry visit a casino.
In one game, they have a 1/5 probability of winning $100 and 1/2 probability of losing $50.
They also have a chance of no win / no loss.
What is the most likely amount of money they will win (or lose) at the end of 100 games?
Gambling is bad!
Gerry and Jane are younger than 25.
The probability that two randomly chosen people in their city are younger than 25 is 25%.
What is the probability that 4 randomly chosen people from the city are younger than 25?
Alex received a 60 on his essay and an 80 on his final.
He got a 90 on class participation.
The essay counts as 30% of his grade.
Class participation counts as 20% of his grade.
What is his grade?
Everyone in tenth grade classes voted on a motion.
Sixty percent of the girls voted YES.
Forty percent of the boys voted YES.
The motion passes if over 50% of the votes are YES.
In which class was the motion rejected?
John has an average of 87 on his two math exams.
To earn an A, he must have a 90 average.
What is the lowest grade he must earn on the next exam to raise his average to 90?
Anna has 3 bags of marbles.
The first contains 5 white marbles and 5 green marbles.
The second contains 2 white marbles and 8 blue marbles.
The third contains 16 white marbles and 4 green marbles.
If she randomly selects a single marble from each bag, what is the probability that all three marbles will be white?
The average (arithmetic mean) of a set of 10 different numbers is 98.
If the numbers 40 and 140 are removed from the set, what is the average of the remaining numbers?
Fifteen numbers have an average of 10.
Five of these numbers have an average of 5, four other numbers have an average of 4, three an average of 3, and two an average of 2.
What is the remaining number?
The average of twelve numbers is 8 and the average of another set of eight numbers is 12.
What is the average of all these numbers?
Bob is rolling two dice and will subtract the second number from the first number.
What result is Bob most likely to roll?
A group of kids with dogs go for a walk.
There are 49 feet and heads altogether.
There are more dogs than kids.
How many dogs are there?
An entrepreneur wants to hire the best person for a position.
He makes a decision immediately after the interview.
Once rejected, an applicant cannot be recalled.
He interviews N randomly chosen people out of 100 applicants, rejects them and records the best score S.
After that, he continues to interview others and stops when the person has a score better than S.
What number N do you recommend to the cruel man?
In a city, there were seven bridges.
There was a tradition to walk and cross over each of the seven bridges.
If a person starts and finishes at the same point, what is the smallest number of crossings the person would have to make?
San Jose Scrabble® Club No. 21 published a three-letter word list that included 1014 English words.
Estimate the percentage of these words compared with all possible three-letter combinations?
This picture shows the first five rows of Pascal's triangle.
What is the sum of the numbers in the 10th row of the triangle?
Forty non-zero positive numbers are written in a row.
The average of the first two numbers is 1.
The average of the second and third numbers is 2.
The average of the third and fourth numbers is 3 and so on.
What is the last number in the row?
John answered 100 4-option multiple-choice questions.
He is sure that he correctly answered 50% of the questions.
In 30% of the questions, he chose the answer among two options, and he answered all other questions by randomly guessing among the four options.
What score does John expect to receive on the exam?
Anna earned an average of 80% on her four exams.
If she never earned more than 90% what is the lowest possible percentage score she could have received in any one of the exams?
It takes 2 cards to build 1 floor of a card house.
It takes 7 cards to build 2 floors of a card house.
It takes 15 cards to build 3 floors of a card house.
How many cards does it take to build a 10-story house?
If we set out by ranks of 10, we will be one short.
We will also be one short if we set out by ranks of 9, 8, 7, 6, 5, 4, 3, and even 2.
Yet there are fewer than 5000 participants.
How many are we?
Source: Marie Berrondom, Mathematical Games, 1983
After a gun is fired in a saloon, 75% of the cowboys have a wounded ear, 80% have a wounded eye, 85% have a wounded arm, and 90% have a wounded leg.
What is the smallest percentage possible of cowboys who have all four wounds?
A chip is placed at the bottom left corner square of a 5 x 5 grid.
The chip is only moved one space upwards (U) or to the right (R). One of these two directions is randomly chosen at each step, for example, by flipping a fair coin.
Find the probability that the chip reaches the center square of the grid in 4 random moves.
Two dice are thrown.
You can see all 12 numbers except 2 faces that are on the table.
All the numbers on the ten visible faces are added.
Which sum of the numbers on the faces is the most probable?
There are 10 girls and 10 boys in a class.
A teacher randomly chooses two students.
What is the probability that they are a boy and a girl?
Dueling Idiots Problem: three idiots participate in a duel.
They shoot at the same time.
If each idiot randomly chooses one of the other two idiots and successfully shoots him, what is the probability that at least one idiot will survive?
A team of three children, Anna, Bill and Cindy, independently answers true-or-false questions.
Anna answers 90% of the questions correctly.
Bill and Cindy answer 50% of the questions correctly.
The answer is accepted as the team's response when the children give the same answer.
If their answers are different, then the result is ignored.
What is the probability that a team response is correct?
I have eight dimes; five are real, and three are fake.
Whenever a real dime is flipped, it comes up heads with a probability of 0.5.
A fake dime comes up heads up with a probability of 5/6.
What is the probability that a randomly chosen coin will come up heads?
Two princes and two princesses are ready to marry.
Both princes are strong, handsome, and rich.
Both princesses are beautiful, cultured, and elegant.
None of them are smelly, vulgar, or unhealthy.
There is nothing to choose between them.
Everybody randomly and independently chooses a partner without telling anyone.
What is the probability that at least one princess chooses a prince that chooses her?
Assume that boys choose girls and girls choose boys.
Six standard dice are rolled and the numbers on the top faces are added together.
What sum is the most probable?
Bobby takes five tiles with the letters of his name and places them into a bag.
He randomly picks one tile after another from the bag, and does not replace them in the bag.
What is the probability that he chooses the tiles in the same order as his name?
Anna has a bag with 90 wafers.
She eats 14 wafers on the first day.
Every day, she eats one wafer less than the day before.
In how many days does she eat all the wafers?
The average of a set of 12 numbers is 1.
If 18, 50 and -20 are added to the set, what is the new average?
99 numbers have an average of 101.
Ninety of these numbers have an average of 100.
What is the average of the other nine numbers?
John arrives at crossroads A from the North.
He makes four moves each time randomly choosing one of three directions.
For example, he could end up at point B.
What is the probability that he finds himself back at point A?
Statistics show that for every 100 babies born in Funny Town, there are 10 more boys than girls.
What is the probability that Mr. Smith's newborn twins are boys?
Alex received a 70 on his essay and a 90 on his final.
He got an 80 on class participation.
The essay counts as 30% of his grade.
Class participation counts as 20% of his grade.
What is his grade?
A box contains at least ten of each of four different types of apples.
I select apples from the box without looking.
How many apples must I draw to be sure of getting at least three of one kind?
Five students put their sandwiches into five paper bags.
The bags are randomly distributed to the students.
What is the probability that exactly 4 students receive the correct bag?
The first person is 100 cm tall.
Each next person is 20% taller than the person before.
Who will be taller than 2 meters first?
Which number could be removed from the set without changing the average (arithmetic mean of all the numbers)?
A city hall contains 165 chairs. The first row has 10 chairs.
Each additional row has one more chair than the row before it.
How many rows are there?
If two dice are rolled 72 times, how many times is the sum of the two top numbers expected to be 12?
A bag contains 6 green, 5 gray, and 4 violet disks.
If 2 disks are drawn at random from the bag, what is the probability that the disks drawn are green?
Five percent of the marbles in a jar are black.
One fourth of the marbles are green.
One half of the marbles are yellow.
The rest are white.
If there are 16 white marbles, how many marbles are in the jar?
Find the natural value k.
What is the value of this infinite series?
What happens to this series as the number of terms increases without limit?
What is the next number in this integer sequence?
Find A. 
Evaluate the infinite series shown.
What is the probability that there are 53 Sundays in a non-leap year?
Find the result of the expression for 
Evaluate this infinite series. 
For the set { 12, 22, 32, 42, ... 102 }, when is the average value a natural number?
In how many ways can I color the six faces of a cube with six different colors?
10385
How many different ways are there from the top to the bottom?
Find X.
Two cards are randomly drawn. 
How many 10-digit numbers can you compose from 0s and 1s? 
What is the simple interest earned on an account of $800 at annual 8% interest for 9 months?
Find X.
Which of the following is closest to 
What is the sum of the first 13 numbers that are evenly divisible by 13?
What time is it 1234 minutes after 1:23 a.m.?
What is the probability of not getting blue in two runs?
Every day, a stock price of $444 is either increased or decreased by 4%. 
Which year has had the greatest number of Roman numerals in it?
If you are a redhead you have more than the average amount of hair. 
What is the average of the smaller of two random numbers from 0 to 1?
The sum of N consecutive integer numbers is 2090.