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Questions tagged [probability]

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

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24 votes
4 answers
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The Monty Hall problem

I'm amazed that, although well known, this problem hasn't been posted yet. Monty Hall You are on a game show. The host of the show shows you three doors. He tells you that behind one of them, is a ...
Cruncher's user avatar
  • 719
6 votes
7 answers
8k views

Help understanding the solution to a "shooting puzzle"

I need help understanding the solution to the following puzzle: Puzzle: There is a "triangular" duel between the three shooters. Everyone shoots one by one, can shoot once and any person he wants. ...
makaed's user avatar
  • 199
5 votes
2 answers
2k views

A modified version of apples and oranges logic

Question: You have 3 baskets, one with apples, one with oranges and one with both apples and oranges mixed. Each basket is closed and is labeled with ‘Apples’, ‘Oranges’ and ‘Apples and Oranges’. ...
AbsoluteSith's user avatar
16 votes
3 answers
113k views

If you choose an answer to this question at random, what is the chance you will be correct? [duplicate]

If you choose an answer to this question at random, what is the chance you will be correct? A) 25% B) 50% C) 60% D) 25% (source: twimg.com)
Larry OBrien's user avatar
38 votes
3 answers
6k views

Do better than chance

You and an opponent are playing the following game: Your opponent picks two different real numbers, however he wants to, and writes them on two cards, which he presents to you face down You choose ...
Ben Aaronson's user avatar
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18 votes
11 answers
4k views

How can the "Boy or Girl" paradox be worded unambiguously?

The first time I encountered the Boy or Girl paradox I was struck by how ambiguous all of the formulations in English are. As Wikipedia notes: Gardner argued that a "failure to specify the ...
Jon Ericson's user avatar
15 votes
2 answers
1k views

Crazy Dice - Unusual Dice with Usual Sum

Can you design two six-sided dice (different from the standard ones), where each face has a nonzero number of dots, so that the probability distribution of their total is the same as for two standard ...
Mike Earnest's user avatar
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13 votes
2 answers
1k views

How to survive, with statistics

You are kidnapped and your offender plays a game with you. In front of you are 2 boxes, containing a total of 50 white balls and 50 black balls. The kidnapper will pull out a ball from one of the two ...
hetzi's user avatar
  • 305
11 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
8 votes
4 answers
1k views

Social Engineering (The Birthrate Problem)

The distant country of Toomalia has a birthrate of 67 boys for every 33 girls. In an effort to restore a 50:50 gender ratio in the country, Toomalian policymakers institute a mandatory new social ...
COTO's user avatar
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52 votes
7 answers
21k views

Does the drunk man fall off the cliff? (a random walk problem)

A drunk man stands with a cliff one step to his left. He takes steps randomly left and right. Each step has probability $p$ of going left and probability $q=1-p$ of going right. Each step is the same ...
DanielSank's user avatar
50 votes
7 answers
11k views

How many tries to roll a 6?

Suppose you roll a (fair, 6-sided, perfectly ordinary) die repeatedly until you roll a 6. As is well known, the expected (i.e., long-term average over many trials) number of rolls required is 6. Now ...
27 votes
10 answers
4k views

Make 2 dice out of 3 dice

This question was first posted by standupmaths on youtube: https://www.youtube.com/watch?v=xHh0ui5mi_E He already got some answers but I think this community can probably find a better answer. ...
Sumyrda - remember Monica's user avatar
16 votes
4 answers
6k views

The Mexican Standoff

There are three cowboys in a Mexican standoff against each other, named Juan, José, and Jorge. Juan is the straightest shooter in the West, and can hit his target 100% of the time. José, who has a ...
user avatar
11 votes
2 answers
861 views

What is the probability that all the bullets will be destroyed?

A gun shots bullets with different speed each second. The bullets form a straight line. The bullets keep their speed (not getting slower or faster). But if 2 of the bullets collide, 2 of them are ...
Jamal Senjaya's user avatar
9 votes
3 answers
2k views

A first choice die

You and your friend have met to play a game of dice. This is a very simple game where both players roll a die at the same time, and the player with the lower score pays a coin to the other player. In ...
GOTO 0's user avatar
  • 13.5k
6 votes
1 answer
1k views

Odds of duplicate birthdays

I found the following problem in an number of places online: John once bet a fellow gambler that two of the first thirty persons they met and spoke to would prove to have the same birthday. Strong in ...
Xynariz's user avatar
  • 3,620
3 votes
10 answers
7k views

Multiple Choice Puzzle

What are the chances of getting this correct if you pick at random? 1/4 1/2 1/3 1/4 You are not allowed to add more answers to this list Note; This DOES have a correct, demonstrable answer!
warspyking's user avatar
  • 14.5k
93 votes
9 answers
14k views

Coin Flipping Game with the Devil

You die, and awake in Hell. Satan awaits you, and has prepared a curious game. He has arranged $n$ quarters in a line, going in the east/west direction. He placed the coins at the ends tails up, and ...
Mike Earnest's user avatar
  • 32.5k
62 votes
3 answers
7k views

Left coin, right coin, last coin?

In this game, all players start sitting at a round table, each with a coin in their hand. At every turn, all players toss their coins together and put them on the table in front of them. Then each ...
GOTO 0's user avatar
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43 votes
3 answers
4k views

Relabeling two 20-sided dice without changing their total

Usually, 20-sided dice are labeled with the numbers 1 to 20. When you roll two of these, their sum is a random number between 2 and 40. The total 21 is most likely to occur, while 2 and 40 are the ...
Mike Earnest's user avatar
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42 votes
1 answer
3k views

Crippled King Crossing a Canyon

A chess king has been injured in battle against an evil wizard, and can no longer move northeast or southwest. This king is on the North rim of a canyon, and must flee to safety on the South rim. ...
Mike Earnest's user avatar
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33 votes
3 answers
3k views

A Strangely Familiar Probability Problem

You're taking the final exam for your Introduction to Probability and Statistics course, and once again you've failed to study (you were up all night solving online brainteasers). The exam is ...
COTO's user avatar
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27 votes
19 answers
9k views

Ant walking on a number line

An ant walks on a number line, starting at location $x=7$. Each second, it randomly moves one space either left ($-1$) or right ($+1$), with equal chance. At $x=0$ and $x=10$ are drops of honey; the ...
xnor's user avatar
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24 votes
2 answers
3k views

A Bridge Problem

You are on the North Shore of a river, and you want to get to the South Shore. In the river are 6 islands and 13 bridges connecting the islands and shores. However, a storm gives a 50% chance to each ...
wyskoj's user avatar
  • 405
20 votes
7 answers
2k views

Triangle in a circle

Suppose three points are chosen at random in a circle. A triangle is made with these three points as vertices. What's the probability that the triangle contains the origin of the circle? (Although I ...
Simon S's user avatar
  • 351
19 votes
6 answers
3k views

Cooperative guessing game: no incorrect guesses

There are 15 players who will play a cooperative game. They begin by closing their eyes. A referee will place either a black hat or a white hat (chosen by fair coin flip) on each player's head. The ...
Julian Rosen's user avatar
  • 14.3k
15 votes
7 answers
3k views

Loaded Dice with Uniform Sum

A loaded die looks like a regular six-sided die, but instead of each number 1-6 being equally likely, some numbers may be more or less likely than others. Is it possible to have two loaded dice so ...
Mike Earnest's user avatar
  • 32.5k
14 votes
7 answers
6k views

Fair n-sided dice

I have recently bought a fair $10$-sided dice for generating equally distributed random numbers from the range $1,2,\ldots,10$. Let me describe this dice in some more detail. Consider a regular 5-gon $...
Alexis's user avatar
  • 8,312
9 votes
6 answers
2k views

Making two fair dice

You are given two unlabelled standard 6-sided dice. Can you write every number from 1 to 12 on the dice, such that in any given throw there is a 50% chance of the first die being smaller than the ...
Dmitry Kamenetsky's user avatar
8 votes
2 answers
2k views

Making three fair dice

You are given three unlabelled standard 6-sided dice. Can you write every number from 1 to 18 on the dice, such that in any given throw each die has the same chance of being the highest? Here is a ...
Dmitry Kamenetsky's user avatar
7 votes
3 answers
535 views

Guessing game with infinity coin flips

Evan and Ollie are going to play a game as a team. The two friends placed in separate rooms, and then a coin is flipped infinitely many times*. Evan is told the results of the even numbered flips, ...
Mike Earnest's user avatar
  • 32.5k
6 votes
5 answers
595 views

Russian roulette

You are challenged to a game of Russian roulette. Your opponent places two bullets side by side in a six chamber revolver and spins the chamber. She explains that the game is single turn-based and ...
Robert's user avatar
  • 71
5 votes
2 answers
3k views

Probability of three segments forming a triangle

Consider a line segment of length L. This segment is cut in three segments at arbitrary points. What is the probability that these three segments could be rearranged as sides of a triangle?
user3883664's user avatar
5 votes
1 answer
665 views

The Minotaur's Labyrinth

You are trapped in a chamber in the center of the Minotaur's Labyrinth. There are $\mathbf{N}$ tunnels, $\mathbf{m}$ of which lead to safety; the remaining tunnels only lead back to the chamber. Each ...
knrumsey's user avatar
  • 672
41 votes
22 answers
9k views

Aproximating 100 by 6

You're a fan of pen & paper RPGs and have finally arranged a new playing group, recruited amongst your friends from the mathematical faculty. As a GameMaster (GM) you've assumed that all your ...
BmyGuest's user avatar
  • 18.2k
37 votes
1 answer
5k views

Say 100 and win

Our good friends that like games, Alice and Bob, play a game called "Say 100 and win". The rules are: In the beginning they use one regular 6 faced die to get a random number between 1 and 6. Let's ...
Marius's user avatar
  • 18.2k
31 votes
3 answers
4k views

A lonely pawn on the chessboard

Alice and Bob and the referee Conny play the following game with a pawn on a standard $8\times8$ chessboard: In the beginning, Conny places the pawn into the center of a randomly chosen square. All $...
Gamow's user avatar
  • 45.7k
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
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
  • 7,407
23 votes
3 answers
2k views

Gambling in a rigged casino

A well-known crooked casino is offering the following bet with a pair of (identically looking and indistinguishable) dice that both show the numbers $1,2,3,4,5,6$ on their faces. The first die ...
Gamow's user avatar
  • 45.7k
22 votes
3 answers
3k views

Fair results with unfair die

You are given a 6-sided die.The die is biased, so it is more likely to produce some results than others, but you don't know the exact odds. It is however given that the odds are the same for each roll,...
aaaaaaaaaaaa's user avatar
  • 2,089
21 votes
2 answers
2k views

One Hundred Lockboxes of Wood and Steel

A bit beyond perceptions reach, I sometimes believe I see that life is two locked boxes, each containing the other’s key. ― Piet Hein You have one hundred lockboxes, fifty made of steel, ...
Mike Earnest's user avatar
  • 32.5k
21 votes
5 answers
7k views

Flip a Fair Coin

I found this question and became curious, can anyone tell me the answer and prove it, i know it seems fairly simple but just thought an explanation of this would make an interesting case. Flip a fair ...
skv's user avatar
  • 5,002
20 votes
3 answers
1k views

Shooting Free Throws

Shaq is shooting free throws. He makes his first shot, then misses his second. Confidence is a huge factor in how well Shaq plays. This means that for each subsequent shot, the probability he makes ...
Mike Earnest's user avatar
  • 32.5k
20 votes
6 answers
6k views

What is the expected value of this coin flipping game?

A casino offers the following game: You flip a fair coin repeatedly and record the results on slips of paper, one result per slip. At any time you may decide to stop, at which time one of the slips ...
Julian Rosen's user avatar
  • 14.3k
17 votes
15 answers
6k views

How to get a uniform 1 out of 7 chance, using only a coin and a 6-sided die [closed]

There are seven days in a week. Let's say you want to pick one at random, such that each day has an equal chance of being picked as each other day. In other words, the goal is to obtain a uniform ...
DrZ214's user avatar
  • 297
17 votes
1 answer
2k views

Contagious ants on a stick

Twenty-five ants are scattered on a meter stick. At the same time, they each pick a random direction (east or west) and start marching at 1cm/second. Whenever two ants meet, they turn around. When an ...
Mike Earnest's user avatar
  • 32.5k
15 votes
11 answers
4k views

Monty Hall all over again?

Often, when people ask the Monty Hall problem, they omit some important details, such that the problem becomes ambiguous, or they add more details, such that the problem changes drastically. For ...
Ishamael's user avatar
  • 414
15 votes
8 answers
5k views

Unfair coins at South Park Elementary

Wendy Testaburger and Sally Turner play a game with two unfair coins. A coin flip with Wendy's coin shows head with probability $\frac {1} {100}$. A coin flip with Sally's coin shows head with ...
Alexis's user avatar
  • 8,312