Questions tagged [probability]

A mathematical puzzle whose essential nature involves randomness. Use with [mathematics]

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The Truth And Lies Today: Revisited

A man is only capable of telling the truth four hours a day, and they are either TTFTT or TFTTT (equiprobable). On the other twenty hours, he lies. All hours between 7pm and 6am are night, 7am and ...
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1 vote
1 answer
123 views

The Truth And Lies Today

Suppose a man only tells the truth three hours a day. It is the same every week and it is known that the pattern of the truth hours are TFTFT. (which means he may tell the truth at midnight and 2am ...
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13 votes
2 answers
527 views

Who is most likely to win all the chips?

Alice, Bob and Charlie each have 5 chips. Starting from Alice, they take turns to act in the order of Alice, Bob and Charlie. The $n$-th person to act must take a total of $n$ chips from their ...
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2 votes
2 answers
254 views

I'm not a robot, I'm not a MONKEY

The city of Roboville has been invaded by robots, monkeys, and dancers, who will dance if the beat's funky. Some of the invaders can be two of them (e.g. both monkeys and robots) and some can be all ...
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0 votes
1 answer
155 views

Will Romeo meet Juliet? [duplicate]

Romeo and Juliet try to meet each other everyday at a certain place between 12:00 and 13:00. They arrive at the place at a random time between 12:00 and 13:00 and they wait 15 minutes (but never after ...
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7 votes
1 answer
281 views

Bertrand's Ballot Theorem

A total of X voting papers for candidate A and Y voting papers for candidate B were cast in one section during the election. Where X > Y At the end of election day, the voting papers were counted ...
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  • 155
6 votes
1 answer
224 views

Follow the randomly-bouncing ball

We are going to generate information out of nothing (well, nothing and a bit of something) just by randomly bouncing a ball. The result will be a message you can use to protect humanity. Let's use ...
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1 vote
2 answers
115 views

What's the correct intuition behind the expected number of die rolls until a 6 if all previous are even? [duplicate]

I came across a puzzle on YouTube recently (spoiler). You roll a fair dice until you get a 6. What is the expected number of rolls, including the roll of 6, conditioned on the event that all previous ...
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2 votes
2 answers
343 views

Choose the wine

I based this on a problem from a mathematics presentation, adding a small twist. I did not readily find it here. Your friend comes to dinner and you know he loves to drink Beaujolais. You have 'Cote ...
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2 votes
1 answer
152 views

Hitting twice with different choices

This game of two players has public parameters an integer $n\ge2$, and a probability $p$ with $1/n<p\le1$. E.g. $n=4$, $p=1/3$. In the first phase of a game, a player secretly decides $n$ ...
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2 votes
1 answer
232 views

Probability that there will be no mutual best friendships? [closed]

Here is a problem: There are two groups of n users, 'A' and 'B'. Each user in A is friends with those in B, and vice versa. Each user in A will randomly choose a user in B as their best friend and ...
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12 votes
2 answers
1k views

Can a fair coin be used to simulate an arbitrary weighted coin, with on average only two flips? [duplicate]

This puzzle was told to me by a Stanford PhD student in a graduate course on randomized algorithms. It's really clever/beautiful, and I think makes for a great challenge. You are given a fair coin, ...
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  • 155
4 votes
2 answers
333 views

A simple guessing game

You and your friend play a simple guessing game. Your friend thinks of a secret whole number between 1 and 10, inclusive. If you ask your friend what the number is, the following happens: With 80% ...
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2 votes
2 answers
262 views

Gambling game where you pick how many coins to flip

At a gambling joint, you are invited to be a player in a game with identical unfair but consistent coins flipped simultaneously in each round, with no cost to enter. Each player gets a different ...
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9 votes
1 answer
362 views

Probability that D wins this contest

Problem: $A, B, C, D$ are playing an archery game. They have equal probability of landing their arrow on any spot the target with radius of $R$. Their score is the distance to the center of the target....
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0 votes
2 answers
363 views

A certain 'Monty Hall' variant was played. How many doors were there, and, what was the value of the prize?

Recall the standard Monty Hall scenario where a presenter hides one prize with high value $h$ behind one closed door (e.g., in the classical version: the one high prize is a car), and, equal prizes ...
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15 votes
6 answers
3k views

Three horse race

There are three horses in the race. You know the following information about them: Horse A will finish the race in 50 or 60 seconds with both events being equally likely. Horse B will always finish ...
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0 votes
2 answers
145 views

How can we allocate when we have 150 open slots every day (5 days a week) for those 200 arrivals every day

My question is to solve a very basic problem related to the allocation of slots. Say there are 20 teams with 10 persons in each team. I have 150 open slots every day (5 days a week) for those 20 teams ...
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4 votes
2 answers
530 views

How many seats around the table?

This puzzle is very much inspired by a recent, excellent, one, posted here. If anyone, in particular the author of the other puzzle, prefers this one to disappear, I will immediately do so and I ...
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3 votes
1 answer
113 views

Two pairs of dice with the same probability [duplicate]

Alice has two dice each numbered from 1 to 6. Bob has two dice as well, the numbering is not known, but each face has a positive integer. The numbering on each dice can be different and a number can ...
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4 votes
2 answers
228 views

The Game of MIM

In the game of MIM, each of two players, Alice and Bob, receive a board as shown below: They are then dealt in turns one card from a shuffled set of cards numbered 0 through 9 which they must ...
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25 votes
4 answers
4k views

The "Loop of rope" dilemma

It's a "dilemma" that I encountered in real life problems. I believe that it's not new and is quite easy to explain, but may look puzzling at first glance. Alice and Bob are playing with a ...
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3 votes
1 answer
196 views

What is the most STAMINA-efficient strategy to escape the well?

You are roleplaying as an adventurer under the direction of a sadistic DM who has just thrown your character down a well. In order to escape, you have unlimited chances at a STAMINA check, difficulty ...
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  • 1,758
2 votes
2 answers
243 views

Extension to the coin flipping puzzle

This is an extension to this great puzzle: Coin Flipping Puzzle. The main difference is we now have ternary sequences instead of binary sequences. Two criminals $A$ and $B$, were recently captured and ...
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1 vote
1 answer
232 views

Optimal parking at a concert

You are going to a concert. The road to the concert has 100 car parking spaces arranged in a straight line. The spaces are numbered from 1 to 100, where space 1 is in the beginning of your journey and ...
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6 votes
1 answer
843 views

Three cups and a ball

I am walking down the streets of Paris when a street performer approaches me. He has a row of 3 cups facing down on a table. He places a ball under the left cup. He proceeds to swap pairs of cups, ...
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10 votes
2 answers
518 views

How should you distribute the fighting power?

Alice and Bob both have N warriors under their command, numbered 1~N, and $1$ point of fighting power at their disposal. Before the game, they privately distribute the power between their warriors. ...
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0 votes
2 answers
167 views

Odds of there being two solutions to a word puzzle [closed]

What are the odds of there being two solutions to, say, a cryptoquote? And is anyone aware of an example of such?
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  • 301
6 votes
2 answers
454 views

A three-way bet

Alice, Bob and Carole are playing a betting game. At the beginning, they have 20 chips in the common pot. They move alternatively in the order of Alice, Bob and Carole. In their move, a player ...
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5 votes
3 answers
713 views

Number of cards to be pulled

Ten cards, numbered 1-10, are randomly dealt to 2 players, A and B, such that each player gets 5 cards. Before the cards are dealt, player A will say a number N. After the cards are dealt, A will ...
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12 votes
1 answer
404 views

How should you bet?

Alice and Bob are playing a betting game. Each has 20 chips to start with. They move alternatively starting with Alice. In their moves, they name a number $n$ and toss a fair coin: if head, they get $...
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9 votes
9 answers
3k views

How many gold coins can you extract from the billionaire?

An eccentric billionaire plays a game with you. She has an urn with 100 gold coins. Each time, you can take any number of coins from the urn. If you take n coins, she will flip a fair coin. If head, ...
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  • 4,883
10 votes
4 answers
2k views

Simulating a biased coin with an unbiased one

You are given an unbiased fair coin, ie., a coin that produces heads with probability 0.5. Can you use this coin to simulate a biased coin that produces heads with some given probability $p$ ? This is ...
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26 votes
7 answers
4k views

Simulating an unbiased coin with a biased one

You are given a biased coin. With probability $p$ it produces heads, where $p$ is in the range $(0,1)$. How can you use this coin to simulate an unbiased fair coin, ie., a coin that produces heads ...
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4 votes
2 answers
445 views

Slow & Fast drivers

Let us define "fastness" as a characteristic of how fast/agressive a driver is. One aspect of this is the (maximum) time between a traffic light turning red and the driver deciding they have ...
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12 votes
3 answers
1k views

Even odds from die with single mark

We want to make a close equivalent of a fair coin throw, by throwing an N-sided fair die T times. The catch: only one side is marked, by a colored disk on that side's center (like the traditional ...
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  • 383
23 votes
4 answers
2k views

Which die should you choose?

You are playing a game with dice. You have a grid with 13 cells shown below. A token starts in the left-most cell (start). Each turn you roll an $n$-sided die and get a number $x$ between $1$ and $n$, ...
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12 votes
3 answers
829 views

Outmaneuver your opponents in the duel

A, B and C are in a three way duel. Starting with A, they rotate in the order of A-B-C, each firing one shot at a time. They stand close to one another, so that each can kill one of the others or ...
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-1 votes
1 answer
112 views

Blindly choosing a ball from a bag [closed]

You are blindfolded. You are handed a bag and told that there are three balls and two cubes, all of which are the same size and are equally likely to be chosen. You blindly choose one of the items ...
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6 votes
2 answers
711 views

Balancing the marble game (Red marbles are OP, please nerf)

Ash and Bree are playing a simple game of chance: They fill a bag with small marbles coloured red or blue. They take turns to draw a marble from the bag without looking, then: If Ash draws a red ...
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  • 469
2 votes
2 answers
510 views

Lightbulbs in a 3×3 square

Suppose we have a $3\times 3$ arrangement of lightbulbs and we switch them on/off randomly (probability $½$). What is the probability the no adjacent bulbs are on? My attempt was: Let $1= $ on and $0 =...
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  • 61
1 vote
1 answer
177 views

Variation of 100 prisoners [duplicate]

There are 100 prisoners, each prisoner is given a number between 1 and 100 without knowing which number, the number can repeat and not all of the numbers must be given. Each prisoner in his turn can ...
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7 votes
9 answers
3k views

"We accept 20% of candidates and I'm 90% good at my job..." [closed]

There is a famous investment firm where a partner was known for having candidates walk into the room without a greeting, sit across from him at a huge desk, and start right away by asking: "We ...
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1 vote
2 answers
251 views

Anne and Bernard's marbles

Anne and Bernard have between them 100 marbles, Anne the most. Some in each bag of their marbles are red, the rest, though fewer, are blue. If Anne draws at random two marbles from her bag of marbles, ...
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7 votes
1 answer
261 views

Stochastic Taxicab Path

A city's roadworks is laid out as a perfectly rectangular tiling. A commuter within this city has to travel to work a distance of 17 blocks east and 7 blocks north each day, and tries to take the same ...
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  • 1,758
0 votes
2 answers
104 views

Car registration similarity [closed]

In my city, car registration plates contain 3 numbers (0 to 9) and 3 letters (A to Z). Today I've noticed that my neighbour's car has the same registration as my car except for one character. Should I ...
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9 votes
5 answers
477 views

The Two Envelope Game Show

This is an original riddle I made, heavily inspired by the Two Envelope problem. I have my answer to this problem, but (full disclosure) even I’m not 100% sure I’ve got it right. I post it half ...
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  • 410
19 votes
1 answer
865 views

Honeybee Hangover

A drunken honeybee lands on a completely random hexagon of a large triangular section (depicted below) of its hive, and then every second afterwards, takes a step to a completely random adjacent ...
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  • 1,758
18 votes
1 answer
587 views

Lap Theory Optimal Betting Game

Three prisoners are seated at a table. Each of them has a mobile phone on their lap, and they are not allowed to look at anyone else’s phone (and obviously no other form of communication is allowed). ...
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  • 6,569
9 votes
2 answers
453 views

Three way duel, which guns to choose?

Alice, Bob and Carole are involved in a game of three way duel. They take turns to shoot in the order of Alice-Bob-Carole-Alice-... until only one surviver is left standing. The rule is very simple: ...
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