Questions tagged [probability]

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

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2 answers
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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 ...
Dr Xorile's user avatar
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2 votes
2 answers
405 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 ...
FirstName LastName's user avatar
2 votes
1 answer
172 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 $n$ ...
fgrieu's user avatar
  • 383
2 votes
1 answer
1k 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 ...
StochasticMan's user avatar
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, ...
weissguy's user avatar
  • 195
4 votes
2 answers
380 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% ...
Dmitry Kamenetsky's user avatar
2 votes
2 answers
299 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 ...
James Waldby - jwpat7's user avatar
9 votes
1 answer
380 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....
Brandon Bolden's user avatar
1 vote
2 answers
444 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 ...
FirstName LastName's user avatar
16 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 ...
Dmitry Kamenetsky's user avatar
0 votes
2 answers
165 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 ...
user3762120's user avatar
4 votes
2 answers
603 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 ...
FirstName LastName's user avatar
3 votes
1 answer
146 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 ...
ThomasL's user avatar
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5 votes
2 answers
911 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 ...
Bernardo Recamán Santos's user avatar
26 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 ...
WhatsUp's user avatar
  • 7,387
3 votes
1 answer
222 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 ...
Feryll's user avatar
  • 2,379
2 votes
2 answers
270 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 ...
Dmitry Kamenetsky's user avatar
1 vote
1 answer
253 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 ...
Dmitry Kamenetsky's user avatar
6 votes
1 answer
1k 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, ...
Dmitry Kamenetsky's user avatar
10 votes
2 answers
554 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. ...
Eric's user avatar
  • 6,498
0 votes
2 answers
177 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?
JoeB's user avatar
  • 301
6 votes
2 answers
504 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 ...
Eric's user avatar
  • 6,498
5 votes
3 answers
768 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 ...
Sagar Chand's user avatar
12 votes
1 answer
462 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 $...
Eric's user avatar
  • 6,498
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, ...
Eric's user avatar
  • 6,498
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 ...
Dmitry Kamenetsky's user avatar
28 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 ...
Dmitry Kamenetsky's user avatar
4 votes
2 answers
458 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 ...
George Menoutis's user avatar
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 ...
fgrieu's user avatar
  • 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 $n$, ...
Dmitry Kamenetsky's user avatar
13 votes
3 answers
896 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 ...
Eric's user avatar
  • 6,498
-1 votes
1 answer
126 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 ...
WarpPrime's user avatar
  • 381
6 votes
2 answers
736 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 ...
Chengarda's user avatar
  • 3,636
2 votes
2 answers
671 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 =...
Lac's user avatar
  • 61
1 vote
1 answer
253 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 ...
asd's user avatar
  • 11
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 ...
ZenBalance's user avatar
1 vote
2 answers
597 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, ...
Bernardo Recamán Santos's user avatar
7 votes
1 answer
296 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 ...
Feryll's user avatar
  • 2,379
0 votes
2 answers
111 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 ...
Dmitry Kamenetsky's user avatar
10 votes
3 answers
751 views

Optimal refueling strategy

On the way home from work, I want to refuel my car. There are 5 petrol stations along the way that all have a different price for petrol. I know that the prices will be in the range [A, B], chosen ...
Dmitry Kamenetsky's user avatar
9 votes
5 answers
552 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 ...
windblade's user avatar
  • 410
19 votes
1 answer
898 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 ...
Feryll's user avatar
  • 2,379
18 votes
1 answer
653 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). ...
happystar's user avatar
  • 7,594
9 votes
2 answers
560 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: ...
Eric's user avatar
  • 6,498
11 votes
1 answer
888 views

Lawbreaking by the numbers

Special Agent Benford has been stymied by a third unsolved case of miscreant creative accounting and will voluntarily retire from the Fraudulent Numbers Task Force. Benford’s ability to detect ...
humn's user avatar
  • 21.9k
11 votes
5 answers
1k views

Two player Monty Hall variation

Alice and Bob are going on a game show as partners. In this game, there are three doors, which contain a car, the car keys, and a goat, which are arranged randomly and secretly behind the doors. Alice ...
Nuclear Hoagie's user avatar
6 votes
4 answers
1k views

A bet with your friend

James, your friend, has invited you to bet with him. He has a fair die, with $3$ faces showing $0$ and $3$ faces showing $1$. You pay him $\\\$70$. He throws the die $15$ times, and records the sum of ...
Culver Kwan's user avatar
  • 5,600
-5 votes
1 answer
175 views

You went to the casino again [closed]

You went to the casino again and saw a cardboard, with the following written on it: There are $\color{brown}{\text{dirt}}$ coins. Every time, you pay \$$10$. Then the coins will be thrown, and you ...
Culver Kwan's user avatar
  • 5,600
1 vote
3 answers
219 views

Random sum adds to 1000 [closed]

A random whole number is generated between 1 and 1000 inclusive, and added to a sum. A new randomly generated whole number is generated and added to the sum until the total sum is greater than or ...
Michael Moschella's user avatar
6 votes
5 answers
1k views

Probability of three identical digits on a four-digit digital clock?

On a digital clock showing 00:00 to 23:59, what is the probability of having exactly three identical digits?
ThomasL's user avatar
  • 12.2k

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