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Questions tagged [probability]

A mathematical puzzle whose essential nature involves randomness.

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14
votes
13answers
5k 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 ...
10
votes
4answers
1k views

Rocks and Lever (Game) - Goofspiel

I am not sure if there is a name associated with this game but I do not know of one. There was this 2 player game I had once played, and I am wondering if there is a winning strategy. The game is as ...
53
votes
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 ...
11
votes
3answers
1k views

How long until all balls are the same colour?

You have a bag containing $n$ balls of $n$ different colours. Select 2 balls randomly (in order, so there's a first and a second), repaint the first in the same colour as the second, and replace both ...
13
votes
3answers
949 views

Hat Guessing Game in Vegas

A team of 10 people are going to play a cooperative game. While their eyes are closed, a pit boss will place either a red or green hat on each of their heads, chosen by fair coin flip. They then all ...
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 ...
4
votes
4answers
417 views

Queue in the bank

Six men are queuing in the bank when the clerk decides to take a break for 10 minutes! Before leaving, he gives to the six men a numbered ticket, in the same order of the queue (ticket #1 to the first ...
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 ...
4
votes
5answers
1k views

Deserted Island [duplicate]

You are stranded on a deserted island with little to no food sources. By your guesses, you will only be able to survive another 10 days. Your only real chance of survival is getting rescued by a boat. ...
14
votes
5answers
2k views

Who's favored in this dice game?

Alyson and Bethany decide to play a game involving a (standard six-sided) die. Alyson goes first, and rolls the die: if she rolls a 6, then she wins. Otherwise, Bethany takes the die and rolls until ...
2
votes
4answers
1k views

4 player card game, who has advantage? [closed]

Edit for "off topic" "correction". I don't know why this question was considered "off topic". It seemed to draw a lot of attention. The "puzzle" was to figure out which person has the advantage ...
10
votes
3answers
827 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....
1
vote
2answers
179 views

Fumbling with a question already posted

This question, has me stumped. Looking at the first version, how there's two lines of thought that go around in my head, both (seemingly) justifiable. 1 : 'One of them is a girl' implies 'Both ...
11
votes
1answer
571 views

Ants on a Hula Hoop

There are $24$ ants scattered around a hula hoop of circumference $3$ meters . They each randomly, and independently, choose to face clockwise or counterclockwise, and then simultaneously start ...
2
votes
5answers
799 views

Another look at the two children riddle

Inspired by this question about the Monty Hall problem, here's a deeper look at another well-known counter-intuitive probability problem which is often stated in a way that leaves subtle ambiguities: ...
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 ...
5
votes
0answers
131 views

Compare two numbers by looking at one [duplicate]

Alice takes two envelopes and puts some number of dollar bills in each one. These can be any positive whole numbers (we're putting realism aside), but they cannot be the same number. Bob chooses one ...
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 ...
6
votes
5answers
2k views

How to maximize your wages?

You're trying to work out how to maximize the wages you receive in the month of February (in a non-leap year). You must work at least 5 days a week and at least 5 hours a day, but you can also work on ...
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 ...
4
votes
2answers
293 views

Ant walking on a number line 2

I'm going to raise the difficulty of the original question one dimension, so maybe a refresher will be good... Link: Ant walking on a number line I just mentioned i will raise it by one dimension... ...
8
votes
8answers
592 views

Five Haberdashers

This is yet another hat puzzle, but not one of the usual ones! Key differences/commonalities between other hat puzzles are emphasized. Five friends are going to go on a game show. When the game ...
13
votes
4answers
1k 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 ...
7
votes
2answers
1k views

Clash of arrows

During the last country fair, you have seen a weird arrow-shooter machine. Its shape is a regular polygon with $N$ sides of length $1$ and it fires one bolt from each vertex at the same time. The ...
9
votes
1answer
515 views

Expected number of squares in a 20x10 grid with 1% chance of removing each edge

Suppose I make a 20x10 grid out of match sticks, using 21 columns of 10 match sticks and 11 rows of 20 match sticks: ...
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 ...
-2
votes
2answers
202 views

It rains in Gadyukino

It rains in Gadyukino village every other day on average. Assume the probability of rain on any given day is 50% and independent. A local weather forecaster has the sole duty every day of predicting ...
3
votes
15answers
9k views

Three doors, one question

A prisoner wakes up in an empty room (being seated on the floor against the wall). There are three solid doors on three of the room's walls: one to his left, one in the middle in front of him and one ...
8
votes
1answer
263 views

Mission: Impossible 2

Ethan Hunt is tied to a chair and can't get up. The super-villain Matt Fillimore stands in front of him. There's a gun. There's the barrel of the gun, six chambers, all empty. Fillimore is putting two ...
0
votes
0answers
122 views

duel of 3 people shooting [duplicate]

$3$ people $A,B,C$ are shooting each other. The chance that $A$ will hit in a single shot is $0.2$, The chance for $B$ to hit is $0.3$ and for $C$ is $0.1$. If one hit a person, he will continue to ...
9
votes
6answers
2k views

Two men for one gold coin

Two men find an old gold coin and want to have a coin toss with it to decide who gets it. The only problem is the coin is heavier on one side so it comes up heads more than tails. What is a fair way ...
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 ...
6
votes
2answers
399 views

Jewels in the box

Professor Egghead participates today in a local game show. There are six jewels (a diamond, an emerald, a moonstone, a ruby, a sapphire, and a topaz), and there are six boxes that carry the numbers 1,...
1
vote
1answer
367 views

Probability in chess tournament

In a 4 round chess tournament with 16 players (where the loser of each two player match is eliminated and the winner moves on to the next round), the pairings for all matches are decided randomly. The ...
7
votes
9answers
6k views

Card probability problem [duplicate]

You have three cards in a hat. One is white on both sides, one is red on both sides, the third is white on one side and red on the other. You pull a card out of the hat and lay it down. The face is ...
-2
votes
1answer
301 views

The Programmers Bot!

Dr. Probot is a famous inventor, known for many creations, his most recent being "Bot". Bot is the very first working robot, he has 3 very basic instructions: Left Right Self Destruct Left: Rotate ...
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 ...
13
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 $...
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 ...
8
votes
3answers
759 views

The tile-flipping game

The tile-flipping game uses 6 tiles that have a white side and a black side. It starts with the tiles arranged in a line with their white side facing up. $$\circ \circ \circ \circ \circ \circ$$ We ...
12
votes
6answers
614 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 ...
5
votes
4answers
4k views

make a green profit in roulette

Story You meet a rich drunk CEO at a roulette table in Vegas. She explains to you that she has a way of almost guaranteeing a profit at this game by betting on only black. In the process she gets ...
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 ...
12
votes
4answers
1k views

Picking a time to bet

You play a game with a friend: he deals cards face up one a time from a uniformly shuffled deck of 26 black cards and 26 red cards. At some point before all the cards have been dealt, you must call "...
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-...
4
votes
2answers
595 views

Left coin, Right coin, Last coin- A fair version

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 it on the table in front of them. Then each ...
-9
votes
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
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, ...
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 ...
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 ...