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Questions tagged [probability]

A mathematical puzzle whose essential nature involves randomness.

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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 ...
5
votes
7answers
6k 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. ...
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 ...
13
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9answers
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 ...
12
votes
2answers
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 ...
8
votes
4answers
973 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 ...
3
votes
2answers
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’. ...
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. ...
70
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 ...
44
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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 ...
54
votes
3answers
5k 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 ...
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 ...
12
votes
5answers
1k 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 ...
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. ...
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 ...
14
votes
2answers
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 ...
4
votes
2answers
2k 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?
4
votes
1answer
940 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 ...
41
votes
7answers
8k 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 ...
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 ...
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 ...
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 ...
13
votes
4answers
977 views

Cooperative guessing game

A standard deck of 52 cards is shuffled and split into two piles of 26 cards each. Two players cooperate, they must score as many tricks as possible. The first player draw 5 cards from the first ...
16
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6answers
4k 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 ...
27
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 $...
12
votes
7answers
3k 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 $...
10
votes
3answers
813 views

Five pirates splitting gold with a twist

Source: This was posted on the Grey Labyrinth Forums many years ago. The five pirates are back. But this time one of them has the dreaded Zanzibar fever, whose terrible effects are spelled out below....
6
votes
2answers
600 views

Probability of nobody being born in a specific month

Assume 11 persons are born in 11 different months. What is the probability $P(n)$ that none of them is born in the $n$-th month? This would be pretty easy if all months had the same length, then $P(n)...
35
votes
1answer
4k 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 ...
19
votes
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 ...
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 ...
11
votes
2answers
778 views

Expected Number of Tosses Until H,T,T,H

I found a nice little app/game the other day called Probability Puzzles. Apologies to those who, like me, will now obsess until they are all completed. I'm sure there will be many such people on this ...
8
votes
3answers
1k 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 ...
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1answer
1k views

Coins, Dice, and Probability! [closed]

Background: It's time for a little bit of fun, your friends are playing a game and you decide to join them! Gamplay: You each sit down and play on a flat surface. You sit in a circle and each throw ...
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 ...
14
votes
5answers
602 views

Socks which may or may not match

You just won 2016 socks. Some of them are white, some are blue. The color of each sock was randomly chosen, with a 50/50 probability. Is it more probable that the socks can be paired, or that you will ...
14
votes
13answers
4k views

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

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 ...
12
votes
6answers
605 views

Flipping a randomly-lopsided coin

Whenever the Lopsided Mint stamps a coin, its heads-bias $p$ is chosen uniformly at random between $0$ to $1$, which makes it have a probability $p$ of landing heads when flipped. Bob takes a random ...
9
votes
3answers
2k views

Sliding Bolt Puzzle - fastest solution (time-wise)

This is a follow-up question for the Sliding Bolt Puzzle. If you have not solved it yet, you might want to head there first, as the extended discussion of its solution in this question will contain ...
6
votes
2answers
702 views

The Fanatic Fever(twist on the 5 pirates puzzle)

While raiding a tropical island, 6 pirates have an incredibly unlucky day. While they scoured the island in search of treasure, one of them was infected with the Fanatic Fever, a disease which hits ...
6
votes
1answer
387 views

Corollary to “A game with 52 cards”

Inspired by A game with 52 cards Alice and Bob play the following game with two (identical) standard decks of $52$ cards. First Alice secretly arranges one deck of $52$ cards in a long row ...
5
votes
5answers
355 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 ...
5
votes
1answer
526 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 ...
0
votes
3answers
348 views

This Dice Blackjack game - is there Nash equilibrium?

This dice Blackjack is game for two - bettor and dealer. It is played with fair six-sided dice. Bettor: Bettor starts the game. Bettor can roll dice as many times he/she desires while the sum of ...
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7answers
3k 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!
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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
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, ...
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 ...
3
votes
2answers
358 views

The most probable key

You broke into John Doe's house and found the safe. It has a combination lock with 4 dials ranging from 0 to 9 each. ...
2
votes
2answers
347 views

How to beat Count Dracula, part 2

This is a follow-up to "How to beat Count Dracula". Abraham Van Helsing and Jonathan Harker play a game against Count Dracula. The three players agree on the following rules: Count Dracula and ...