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Questions tagged [probability]

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

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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
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65 votes
15 answers
11k views

2 Monkeys on a computer [closed]

(a) Two monkeys are typing capital letters (A-Z) randomly. The first stops typing when the word COCONUT appears as seven successive letters. The second stops typing when TUNOCOC appears; TUNOCOC is ...
anmol porwal's user avatar
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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56 votes
11 answers
7k views

Eccentric Millionaire Probability Paradox

The following is a probability paradox I've been thinking about. It involves Bayes' rule; if you're not familiar, a good starting example is a urn that has a 50% chance of containing one black ball ...
Tyler Seacrest's user avatar
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 ...
48 votes
11 answers
10k views

3 doors, three guards, one stone

You are in a room with three doors. Behind two of them, the darkest pit of hell waits for you to make a mistake. The other door leads to heaven and freedom. Each door is guarded by a guard: Michael, ...
Andrii Chumakov's user avatar
44 votes
4 answers
2k views

Hamlet and the theatre

Hamlet is preparing a play to find out the truth about his father's death. The theatre has $2015$ numbered seats and can contain all the members of King Claudius's court. He gives to each member a ...
leoll2'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
  • 32.5k
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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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
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40 votes
3 answers
4k views

The "M&M Sugar Rush" game

Suppose I have in my possession a incredibly large (almost infinitely large!) bag of M&Ms containing a uniform distribution of the 6 M&M colors. I'm bored, so I decide to play a game: I draw ...
Scott M's user avatar
  • 511
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
  • 2,429
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
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36 votes
5 answers
5k views

My grandfather's socks

My grandfather has a big drawer where he keeps his socks. The drawer contains more than 900 but less than 1000 individual socks. Each of his socks is black or blue, and there are more blue socks than ...
Alexis's user avatar
  • 8,312
34 votes
8 answers
5k views

Monty Hall Revisited: Winning Both Goats!

Suppose you're at a game show, knowing the winning strategy for the classic Monty Hall problem. Your turn comes up and you take the stage. You confidently choose door number (?). Just as you are ...
Tim's user avatar
  • 1,018
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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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
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
  • 27.2k
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
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
25 votes
4 answers
3k views

Digging a tunnel between random locations

Romeo and Juliet are each placed uniformly at random locations on a spherical planet, meaning that each square meter of surface area of the planet is equally likely. Romeo decides to dig a straight-...
Johannes's user avatar
  • 4,400
24 votes
4 answers
4k views

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
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
24 votes
3 answers
3k views

Winning an unfair game

I came across the following interesting problem today: A game consists of a sequence of plays; on each play either you or your opponent scores a point, you with probability 𝑝 < 1/2, he with ...
arshajii's user avatar
  • 2,188
24 votes
1 answer
2k views

Monte Carlo Chess

This is an entry to the 17th fortnightly topic challenge. Chess is an interesting game, but is so unrealistic. In real life, you couldn't tell order your knight to attack an enemy bishop and expect ...
Mike Earnest's user avatar
  • 32.5k
23 votes
3 answers
4k views

bored of eating soup

A man orders spicy noodle and leek soup from a restaurant, but gets bored while eating. When he gets bored, there are exactly 100 noodles in the soup. Because he is bored, he decides to play a game ...
sousben's user avatar
  • 2,586
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
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
  • 2,079
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
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
8 answers
2k views

A Short Dice Puzzle

There are $2$ fair dice: An $11$-sided die, valued from $-5$ to $5$, A $41$-sided die, valued from $-20$ to $20$. You pick one, and I'll take the other one. We roll the dice, and whoever rolls the ...
athin's user avatar
  • 34.2k
21 votes
6 answers
3k views

Distracted Monty Hall

You're about to compete on the standard Monty Hall gameshow. But immediately before the show starts, you overhear the host's wife telling him she wants a divorce. You estimate that with probability 50%...
fblundun's user avatar
  • 1,536
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
20 votes
3 answers
2k views

Winning chance in coins game with fixing

A player plays the following solitaire game. The game consists of as many rounds as are needed to produce a result. They have 20 fair coins, which may in any round be live or fixed; each coin starts ...
Rosie F's user avatar
  • 8,640
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
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
19 votes
1 answer
900 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
  • 2,389
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
18 votes
5 answers
1k views

Blindfold Bingo

Congratulations, you have been selected from among the audience to compete in the fabulous Blindfold Bingo challenge! (cue annoying, yet catchy theme music.) Here you see our perfectly ordinary Bingo ...
Bass's user avatar
  • 77.7k
18 votes
1 answer
655 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
  • 7,664
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
17 votes
4 answers
914 views

Waiting for the gravy at a dinner table

You are sitting at a round dinner table with N other people (i.e. N+1 people all together). One of the other guests is the first to pick up the gravy boat and pour some on their potatoes. It then gets ...
Jaap Scherphuis's user avatar
17 votes
3 answers
675 views

"Musical chair" for clock hands

A colleague and me usually make bets on the time when a simulation is finished. Next time I would suggest the following: At the time the simulation is finished we take a look at the analog clock on ...
A. P.'s user avatar
  • 5,948
17 votes
1 answer
793 views

Two Dice = Nine Cards

I have nine cards, each labeled with a real number (possibly negative or fractional) in a special way. When I randomly pick two of these cards without replacement and add their labels together, the ...
Mike Earnest's user avatar
  • 32.5k
17 votes
6 answers
2k views

Points in a Rectangle

I was asked this question in an interview, so I am not sure what the optimal answer is. But here it goes: Given a rectangle with length l and breadth b, I pick 4 random points inside it. What is the ...
Dipped Bits's user avatar
  • 1,070

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