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# Questions tagged [probability]

A mathematical puzzle whose essential nature involves randomness.

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### Seven cowboys play a coin tossing game [closed]

Seven cowboys play a coin tossing game with a fair coin where each player has a chance of 1/2 of winning. Each of them plays once against all other six. What is the probability that each cowboy wins ...
1k views

### Seven cards in a bowler hat

The game show host puts seven cards with the numbers 1, 2, 4, 8, 16, 32, 64 into a bowler hat. The candidate picks cards up randomly from the hat, one card at a time, and without putting cards back, ...
3k views

### A chess board and a coin! [closed]

You have a standard chessboard, consisting of $64$ squares ($32$ white and $32$ black) each of dimensions $4\times4$ cm, and you have a standard coin of $2$ cm diameter with tail and head. You flip ...
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". ...
397 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 ...
171 views

### A chance to know 52 cards [closed]

This is the Follow up Question to A game with 52 cards. So it requires to understand previous question! I believe it is a bit harder to find! what is the chance to know the positions of all the face-...
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. ...
147 views

### Finding the numbers [duplicate]

I found a nice problem. Here is what it says: Kevin and Peter are given a piece of paper each where they write a number. Their friend John comes up and takes the two pieces of paper and opens ...
400 views

### A game with numbers [closed]

Two people randomly select two different 1-digit numbers (including 0 of course) and calculate their absolute difference. Then They share what they get, What is the probability of having the same ...
562 views

### What is the value of this other coin flipping game?

This is variation of this question: What is the value of this coin flipping game? Imagine that you can play the game in a casino - you are flipping a fair coin. If you get tails, the game continues ...
1k views

### Three Cafes Puzzle

Every morning a couple has breakfast at one of three cafes: Amy's, Bob's, or Cindy's. After eating at Amy's or Bob's, they flip a coin to see which of the other two cafes they'll go to the next ...
162 views

### The Attack on Fort Alpha

I made this puzzle up in 1999 when I was a student. Fort Alpha, on your border territories, was attacked by The Enemy. You don't know if it fell, or repulsed The Enemy's vicious onslaught. A totally ...
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 ...
1k views

### Place 10 Coins Heads Up Blindfolded [duplicate]

You are blindfolded and 10 coins are placed in front of you on table. You are allowed to touch the coins, but can’t tell which way up they are by feel. You are told that there are 5 coins head up, and ...
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 ...
334 views

### Card Passing Game

13 friends are sitting around a table. They shuffle a half-deck of cards, consisting of only the black cards of a standard deck, and deal out two cards to each person. There are then a series of ...
181 views

### Gambling with the gambler [closed]

You are walking on the street and notice a guy gambling with some other guys and you start to watch. The game principle is pretty easy. There is a standard 52-card deck and it is being shuffled every ...
150 views

### Finding a random number in a randomly assigned number in boxes [duplicate]

In a closed room, paper slips labelled $1$ to $100$ are randomly put into boxes also labeled $1$ to $100$. You win a big prize if you can locate a given number (from $1$ to $100$) by opening no more ...
380 views

### Drawing uniformly using a single biased coin

Your task is to draw uniformly from a set of size $N$. You are given a single coin which you can ‘bend’ to control the probability $p\in[0,1]$ of throwing a Head, but you are only allowed a bounded ...
601 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 ...
1k views

### In a Guess The Color game, what's the optimal strategy?

Here's a one-player card game: We begin with a shuffled card deck The player try to guess the color of the card at the top (black or red) If the player guess right, he can keep the card. Else the ...
681 views

1k views

### A coin flip's long tail

I have a (possibly-) unfair coin, which lands heads with probability $p$ and lands tails with probability $1-p$. I toss it twice; the tosses are independent and identically distributed. The ...
4k views

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 ...
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 ...
2k views

### How to cheat at cards

I have a deck of 52 cards on the table in front of me, in random order, each one randomly facing either up or down. I place one joker at each end, each of these again randomly facing either up or down,...
484 views

### One Hundred Lockboxes, Variant

You have one hundred lockboxes, fifty made of steel, fifty made of wood, and a key for each box. A prankster breaks into your house and distributes the keys among the boxes. He places each key in a ...
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, ...
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 ...