Questions tagged [probability]

A mathematical puzzle whose essential nature involves randomness.

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75
votes
8answers
11k 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 ...
61
votes
15answers
9k 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 ...
56
votes
3answers
6k 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 ...
54
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11answers
6k 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 ...
45
votes
7answers
9k 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 ...
45
votes
7answers
13k 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 ...
43
votes
3answers
3k 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 ...
42
votes
11answers
8k views

3 doors, three guards, one stone

You are in a room with three doors. You find out that behind two of these doors, the darkest pit of hell is waiting for you to make a mistake. The other door leads to heaven, where you obviously want ...
41
votes
4answers
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 ...
41
votes
1answer
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. ...
40
votes
3answers
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 ...
39
votes
22answers
8k 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 ...
36
votes
5answers
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 ...
35
votes
1answer
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 ...
34
votes
3answers
5k 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 ...
32
votes
7answers
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 ...
32
votes
3answers
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 ...
28
votes
3answers
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 $...
27
votes
19answers
6k 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 ...
25
votes
10answers
3k 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. ...
23
votes
3answers
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 ...
22
votes
4answers
3k 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 ...
22
votes
3answers
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 ...
21
votes
4answers
2k 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-...
21
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3answers
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 $p < \frac{1}{2}$, he ...
21
votes
1answer
1k 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 ...
20
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2answers
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, ...
19
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5answers
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 ...
19
votes
3answers
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 ...
19
votes
8answers
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 ...
17
votes
5answers
718 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 ...
17
votes
1answer
1k 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 ...
17
votes
6answers
5k 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 ...
17
votes
1answer
586 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 ...
17
votes
6answers
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 ...
17
votes
3answers
2k 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,...
16
votes
6answers
4k views

99 doors to death!

You wake up in a small locked room of a huge castle. There are 100 doors, numbers 1 to 100, and you are to open one of them. Open the correct door, and the entire castle is yours to keep. Open any of ...
16
votes
5answers
2k views

Monty Hall variation

During a train trip I cooked up a semi-autobiographical variation of The Monty Hall problem for people who already know the original. Perhaps it is more of a meta-puzzle than a puzzle, but I hope it ...
16
votes
5answers
2k 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 ...
15
votes
8answers
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 ...
15
votes
5answers
1k views

The Price is Right [closed]

A real life puzzle, yesterday on The Price is Right a contestant played the following game: She could choose 5 cards from a pile where 2 cards were "Car", 11 were "C", 11 were "A" and 11 were "R". ...
15
votes
3answers
2k views

The Sphinx's Dice

"I'm bored of riddles, I'm going to ask you a question about dice instead", says the Sphinx. "Now, I've got this set of Efron Dice here. There's four dice, and they have these values on their faces:" ...
15
votes
4answers
1k views

Another probabilistic coin puzzle

Credit to GOTO0 for the original idea for this puzzle, which is the converse of his puzzle. In this game, all players start sitting at a round table, each with a coin in their hand. At every turn, ...
15
votes
3answers
2k views

A way to beat the system?

I and my friend have always had an interest in gambling. Every week, we each have $£100$ to spare, and we like to bet it on football games at the bookmaker’s. In the long term, we (unsurprisingly) don’...
15
votes
2answers
489 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 ...
14
votes
2answers
5k views

How to simulate one die with three dice?

I do not know if there is a correct solution to this puzzle, but I can't stop thinking about it, so here goes: is it possible to accurately simulate a single, fair, six-sided die using three custom ...
14
votes
11answers
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 ...
14
votes
7answers
2k views

Poker and a card shark

I am playing poker with three friends and from a well-shuffled deck we have each been dealt five cards. I have a hand consisting of the four kings and the two of hearts. Being a poker wizard I know ...
14
votes
10answers
3k 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 ...
14
votes
7answers
2k 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 ...