# Questions tagged [probability]

A mathematical puzzle whose essential nature involves randomness.

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### A hat puzzle involving wizards

Two wizards, Alice and Bob, each have two hats on their heads. Each hat is independently black or white with probability one-half. When the game starts each wizard will see the hats on the other ...
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 ...
288 views

### The four prisoners [closed]

Four prisoners, 1, 2, 3 and 4, are all condemned to death. One of them is at random selected to be pardoned. 1 asks the guard who will be executed. The guard refuses to tell him so instead 1 says '...
47 views

### Knight's Tour Question [duplicate]

A chessboard is an 8 × 8 grid. A knight can move up two squares plus one square to the left or right, or up one plus two squares to the left or right. For a knight in the position shown, determine the ...
151 views

### Halve or diminish, and race to unity! v2

This is follow-up question to Halve or diminish, and race to unity!. Alice and Bob are playing a game. In the beginning, they randomly choose a positive integer between $2$ to $n$ where $n$ is a ...
208 views

### Black socks and green socks [closed]

Some green socks, some black socks, their total is four. Some green socks, some black socks, some socks on the floor. The chance of getting two black ones, the chance of getting two black ones, that ...
615 views

### 5 guards, 5 doors, and probability

1st Guard: Always lies all the time. 2nd Guard: Always tells the truth all the time. 3rd Guard: 75% probability of telling the truth. 4th Guard: 50% probability of telling the truth. 5th Guard: 25%...
797 views

### Best strategy to get all heads/tails from three coins

Your friend has 3 fair coins: a dime, a nickel, and a quarter. She places them on a table behind you. She can arrange then in any pattern of heads and tails as long as they are not all the same. You ...
148 views

### Check-out counter at a supermarket [closed]

A check-out counter at a supermarket will service one customer per unit time if there is anyone in line. Customers arrive at the line: in each unit of time, the probability that a single new customer ...
1k views

### 7 Numbered Balls Game

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 are 7 balls in a pouch and you have 3 balls to draw ...
434 views

### Two Dragons and a Princess

The Quest You are a knight tasked with rescuing the King's daughter. The princess is being held captive by a band of evil dragon pirate monks. You've been debriefed with prior intelligence gathered ...
714 views

### A book with only two letters

Dr. G somehow detests reading novels but he decided to read a one then he naturally got bored and decided to delete all letters except A and B letters in the book. So with this new book which ...
539 views

### Dr. G's Interesting Theorem on Playing Cards

Dr. G has spent his last weekend with analyzing $n$ number of cards which are numbered from $1$ to $n$. He has a fair shuffling machine for his cards and while playing and drawing some cards, he uses ...
267 views

### Optimal solution to a modified poker game

Note: Knowledge of the poker game Texas Hold-em is (probably) needed in order to answer this question. You and a friend (only two people are playing) are playing a standard game of Texas Hold-em but ...
1k views

### Who will win this dice game? [closed]

You are playing a dice game with your friend by rolling two standard dice and recording the sum of the two numbers. You win when two consecutive outcomes are 7. Your friend wins when three consecutive ...
1k views

### Climb the ladder [closed]

You are above a pit of fire and the only way out is a $1000-$ rung escape ladder. You're on the first rung. Every second a standard die rolls. If it's $1$ or $2$ you move down one rung. Otherwise you ...
226 views

### Darts on a dartboard [closed]

I have $x$ darts that I randomly throw on a dartboard with $y$ slots (the v shaped slices like you get when you cut a cake). Now I select a particular slice on the dartboard . What's the probability ...
136 views

### Solitaire wins and losses

Let say a solitaire game can be solved 1 in 10 plays. (A play is defined as shuffling of cards and trying to solve the solitaire). If after each successfully solved game we skip 8 games (we ...
725 views

### What are the chances you lost a nickel?

Yesterday you had 50 coins totaling exactly one US dollar in value. Somehow you lost one of those fifty coins on the way to the vending machine. Assume the likelihood of losing each coin (regardless ...
156 views

### What are the odds that everyone has a partner at a club? [closed]

I badly misread this problem: 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 ...
331 views

### Where the water is hotter

There are two perfectly identical boilers containing the same amount of water, switched on and working in identical environments. Both are setup so that the heating starts when the temperature of ...
604 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 ...
388 views

### A cell in an infinite culture

A single-celled organism is placed into an infinite culture. This cell has the genetic property that it has a $p = 3/4$ chance of splitting into two cells before it dies, and all its children will ...
313 views

### Expected value of a milion dollar (equal?) division

I've heard this riddle a while ago and it is quite interesting. Say I've got two sons and one million dollars to spare. Each boy gets one dollar as a start and from there I give them dollar by dollar ...
195 views

### A taxi was involved in a hit-and-run accident last night [closed]

A taxi was involved in a hit-and-run accident at night. Two taxi companies, the green and the blue, operate in the city. 85% percent of the taxis are green and 15% are blue. A witness identified the ...
417 views

### How to beat Count Dracula, part 3

This is a follow-up to "How to beat Count Dracula" and to "How to beat Count Dracula, part 2". Abraham Van Helsing and Jonathan Harker play a game against Count Dracula. The three players agree on ...
352 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 ...
1k views

### A Lazy Way to Make Two Dice from Many

Recently, I stumbled upon the puzzle: Make Two Dice out of Three, which asks for a solution to a case of the following question: Suppose I roll $n$ indistinguishable dice. From purely the $n$ ...
282 views

### Eight Suspects and a Crazy Executioner

A group of 8 suspects; 3 wearing red shirts, 2 wearing green shirts and 3 wearing blue shirts. Eye-witness claims to have seen 3 people in the scene, the shirt of one of them appeared red, while the ...
3k views

### Deceptive dice game

You must create a set of 6-sided dice and play a game with an opponent. The dice are like regular dice, but you can change the numbers on the dice. For example, regular dice have sides ...
539 views

### Roulette tournament?

You have been invited to play in a French Roulette tournament. The tournament-specific rules are as follows: You start with 100 tokens Play lasts for 5 rounds Minimum bets based on color, low/high (1-...
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’...
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. ...
406 views

### Kangaroo on a square

A kangaroo is jumping around on the corners of the square ABCD. The kangaroo starts at corner A. In each jump it moves to one of the two adjacent corners, with a probability of exactly 1/2 for each. ...
2k views

### Mr. Gardner's dessert

Mr. Gardner is facing the choice of dessert at his favorite restaurant The Erratic Chef. The cook tends to prepare great desserts, but is not very consistent. Mr. Gardner hates it when his choice for ...
305 views

### More coin flipping

After their last game, Alice and Bob decide to have a bit more fun with their fair coin. Bob proposes the following game: considering a flip can come out as heads or tails, they will both predict one ...
485 views

### The asymmetric coin game

After finding a coin in her farm, Alice proposes the following game to Bob: Alice will flip the coin $n$ times. Bob will then flip the coin $2n$ times. If Bob manages to get the same amount of ...
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 ...
244 views

### Coin with two heads [closed]

I was at a job interview and got a question. You have a bag with 100 coins. One of these coins has two heads, other coins are regular. You get one coin from this bag, put it on a table and see head. ...
2k views

### The Good, the Bad and the Ugly

Tuco, Blondie and Sentenza play a game with a single die. Tuco has the first roll, then Blondie, and then Sentenza, and so on. As soon as a player tosses a six, that player drops out of the game and ...
1k views

### A set of lockers

There is a series of 100 lockers in a school. All of them start closed. Students number 1 to 100 flip an even, two sided coin. If the coin lands on heads, they go through each locker corresponding ...
1k views

### No two heads in a row

Suppose I toss a fair coin a 100 times. What is the probability of not getting two heads in a row?
2k views

### Unfair coins at South Park Elementary v2

Following Unfair coins at South Park Elementary Wendy and Sally have another friend, called Timmy. Although Timmy has no coin, he wants to play this game in a fair condition with them. Wendy does the ...
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 ...
947 views

### Domino tile possibilities

I own a normal domino game with 28 tiles. Each tile carries two of the numbers [0, 1, 2, 3, 4, 5, 6] (where 0 represents blank). During cleanup five pieces fell out of the box. Two of them showed ...
889 views

### 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, ...
2k 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 ...
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 ...