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# Questions tagged [probability]

A mathematical puzzle whose essential nature involves randomness. Use with [mathematics]

426 questions
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12 votes
5 answers
2k views

### Unpaired socks in my lap

Today in my laundry basket I had 10 distinct pairs of socks. I repeatedly take a sock randomly from my basket. If it matches a sock in my lap, I pair both up and put them aside, otherwise I put it on ...
13 votes
2 answers
531 views

### The HH vs HT question

My puzzle is based on this tweet (image): Flip a fair coin 100 times—it gives a sequence of heads (H) and tails (T). For each HH in the sequence of flips, Alice gets a point; for each HT, Bob does, ...
17 votes
4 answers
917 views

### Waiting for the gravy at a dinner table

You are sitting at a round dinner table with N other people (i.e. N+1 people all together). One of the other guests is the first to pick up the gravy boat and pour some on their potatoes. It then gets ...
11 votes
2 answers
925 views

### Gold and silver coins in sealed envelopes

Alice has freely chosen to put either a gold coin or a silver coin in each of an infinite sequence of envelopes numbered 1,2,3,... Bob can open any number of envelopes and check the coins within, ...
8 votes
4 answers
1k views

### If the door in Monty Hall is opened by a person without knowledge of where the prize is, and also does not reveal prize, do the odds change?

Here is the quiz variant. After the candidate chooses one door for the first time, and before the master (who knows which door hides the prize) intentionally opens another one to reveal no prize, an ...
1 vote
2 answers
224 views

### Diamond Rush: Back to... um

It is summer in the Pse School Of Logicians, and everyone studying there is dying for a break (they absolutely hate school, and can do anything to get out of it). And because the teachers there are ...
20 votes
3 answers
2k views

### Winning chance in coins game with fixing

A player plays the following solitaire game. The game consists of as many rounds as are needed to produce a result. They have 20 fair coins, which may in any round be live or fixed; each coin starts ...
5 votes
4 answers
733 views

### The Alien Snails Experiment

Fictional Background Story (The Experiment): Two extremely fast and intelligent alien snails are randomly placed on two different points on an infinite plane, as an experiment by the cruel ruler of ...
3 votes
2 answers
435 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 ...
9 votes
2 answers
833 views

### Help! I lost my marble(s)!

Tomorrow is the annual meetup for the Northwest Region Marble Lovers Association. The day's festivities will include various marble games, admiring collections of marbles, a guest speaker who will ...
10 votes
3 answers
756 views

### Optimal refueling strategy

On the way home from work, I want to refuel my car. There are 5 petrol stations along the way that all have a different price for petrol. I know that the prices will be in the range [A, B], chosen ...
2 votes
0 answers
135 views

### Picasso at the art school [closed]

Picasso is preparing for his fine art exam. At the exam he will be shown exactly (a replica of) one of the following four paintings chosen uniformly at random: Mona Lisa The Last Supper The Starry ...
1 vote
0 answers
152 views

### Impossible Minesweeper?

I was playing the game Minesweeper earlier today and came across the following situation: (Look at the bottom-middle section of the gameboard.) It seems to me that either of the two diagonal pathways ...
48 votes
11 answers
10k views

### 3 doors, three guards, one stone

You are in a room with three doors. Behind two of them, the darkest pit of hell waits for you to make a mistake. The other door leads to heaven and freedom. Each door is guarded by a guard: Michael, ...
7 votes
2 answers
424 views

### Does this random sequence contain the number 1?

Randomly choose a number from 1 to 3 and call it $a_1$; Randomly choose a number from 1 to 3$a_1$ and call it $a_2$; Randomly choose a number from 1 to 3$a_2$ and call it $a_3$; $\cdots$ Repeat this ...
-1 votes
1 answer
158 views

### How many pencils are there? [closed]

Four pencils are chosen (without replacement) from a pencil case, two of which are blunt. The probability that both blunt pencils are chosen is twice the probability that neither is chosen. How many ...
4 votes
6 answers
5k views

### What is the probability that your life will have lasted for 100 years once you die?

You are in a world where exactly 90% of all people live for exactly 3 years, and exactly 10% of all people live for exactly 100 years. Aside from what I mention here there is no information that can ...
4 votes
1 answer
374 views

### The Monty Hall loot box

Everyone sufficiently competent in probability knows that in The Monty Hall problem as most commonly presented, switching doors wins you the car $\frac23$ of the time. I have come up with this ...
19 votes
6 answers
3k 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 ...
2 votes
3 answers
423 views

### Button multi arm bandit problem

Let's say you have the following buttons to press, labeled a - j, that would earn you the following amounts of money: a: \$6 b: \$9 c: \$15 d: \$26 e: \$45 f: \$78 g: \$136 h: \$416 i: \$728 j: \$...
6 votes
4 answers
947 views

### Was Humpty Dumpty right?

Raymond Smullyan's What is the Name of This Book? contains a puzzle which I'll paraphrase here: Of the identical twins Tweedledum and Tweedledee, one of them lies on Mondays, Tuesdays and Wednesdays ...
0 votes
1 answer
571 views

### What is the probability I can't bike despite good weather? [closed]

I am trying to solve this puzzle: I have a total of two bikes that I keep at work or at home. I always bike to and from my work unless it rains, in which case I take a taxi (and don't bring my bike). ...
12 votes
8 answers
2k views

### frog on a number line

A Frog is at C. The purple numbers are the probability that the frog jumps in that direction when it is at certain place. The frog stops at A or E. What is the probability that the frog stops at A? ...
0 votes
2 answers
267 views

### Expected number of steps [closed]

There are N cars that are parked in N parking spots numbered from 1 to N. After the Nth parking spot, there are N more parking spots numbered from N+1 to 2N. At each step, a car is selected randomly ...
6 votes
3 answers
730 views

### The Halftime Hustle

In a soccer stadium restroom there are 100 stalls (cubicles), arranged in a straight line. Stalls are numbered 1, 2, 3,..., 100, with 1 being closest to the main exit of the restroom. Unfortunately, ...
7 votes
3 answers
519 views

### Colliding Bullets again

Here's a colliding bullets problem of my own devising that's different from previous versions. Every second a gun has a 60% chance of firing a bullet in a straight line. After 10 seconds there would ...
16 votes
4 answers
6k views

### The Mexican Standoff

There are three cowboys in a Mexican standoff against each other, named Juan, José, and Jorge. Juan is the straightest shooter in the West, and can hit his target 100% of the time. José, who has a ...
24 votes
1 answer
2k 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 ...
3 votes
1 answer
232 views

### Expected power consumed

Suppose you have an 8x8 grid of electric bulbs. So you have 8 rows and 8 columns. Now you can either switch on and supply power to entire row(or column), or switch it off and supply no power at all. A ...
21 votes
6 answers
3k views

### Distracted Monty Hall

You're about to compete on the standard Monty Hall gameshow. But immediately before the show starts, you overhear the host's wife telling him she wants a divorce. You estimate that with probability 50%...
24 votes
2 answers
3k views

### A Bridge Problem

You are on the North Shore of a river, and you want to get to the South Shore. In the river are 6 islands and 13 bridges connecting the islands and shores. However, a storm gives a 50% chance to each ...
2 votes
2 answers
787 views

### Replacing Untold Billions

First: Assume Earth's population is 8,037,860,611 and they have ID numbers 1 through 8,037,860,611 You make a set of cards numbered 1 through 8,037,860,611 A robot goes to each person in turn and ...
7 votes
1 answer
277 views

### Intransitive urns

N balls with numbers 1,2,3...N are filled arbitrarily into 3 urns A,B and C - but no urn may remain empty. We write P(A,B) for the probability that a ball chosen randomly from urn A shows a higher ...
5 votes
2 answers
257 views

### Making fair dice

Dmitry Kamenetsky asked the following two questions: Making two fair dice Making three fair dice My question is a generalization of Dmitry‘s questions. You are given N unlabelled standard 6-sided dice....
8 votes
2 answers
2k views

### Making three fair dice

You are given three unlabelled standard 6-sided dice. Can you write every number from 1 to 18 on the dice, such that in any given throw each die has the same chance of being the highest? Here is a ...
9 votes
6 answers
2k views

### Making two fair dice

You are given two unlabelled standard 6-sided dice. Can you write every number from 1 to 12 on the dice, such that in any given throw there is a 50% chance of the first die being smaller than the ...
8 votes
1 answer
447 views

### Stickered Dice Game

You and your friend play a game. To begin with, each of you has an unlabelled die. On the table laid in front of you are 12 stickers labelled 1-12. The first player chooses one of the stickers and ...
5 votes
1 answer
434 views

### Measuring distance with steps

I am trying to measure 100m for a children's foot race, but I forgot to bring my measuring wheel. I know that each of my steps is between 0.9m and 1.1m (sampled at uniform). I can only take full steps ...
2 votes
1 answer
253 views

### The minimalist game master [closed]

A game master tells his party of players that he has converted to the school of minimalism, and is going to do away with all of his dice in the campaigns they play from now on. In the place of dice, a ...
4 votes
2 answers
2k views

### A prisoner and two dice

A prisoner is offered the following deal. They will be given a pair of standard 6-sided dice to roll. If they can roll a total of 42 or more without rolling a double then they will be set free. ...
2 votes
1 answer
178 views

### Grabbing enough M&Ms to be 80% confident all colors are represented [closed]

Assume you have an arbitrarily large jar of M&Ms, with six different colors of M&Ms present in equal quantities. If you reach into the jar and grab a fistful (what we'll call a "pull"...
-2 votes
1 answer
158 views

### How many earrings do you have to pick up? [closed]

I'm having trouble solving a puzzle in Swedish. I've translated it below: How many earrings do you have to pick up to be sure to get a pair of the same in a mix of earrings where "favorable ...
4 votes
2 answers
529 views

### Prisoners, apples & oranges

You are in a group of sixty prisoners, and the warden has a game to play with you. In a room there are sixty boxes. Each box can contain either two apples, two oranges or one of each, but you don't ...
6 votes
3 answers
1k views

### A series of rock-paper-scissors games

You are playing a series of 100 games of rock-paper-scissors^ against your friend. From your past games, you know that he will choose exactly 40 rocks, 35 papers and 25 scissors in some random order. ...
7 votes
1 answer
349 views

### Follow the randomly-bouncing ball

We are going to generate information out of nothing (well, nothing and a bit of something) just by randomly bouncing a ball. The result will be a message you can use to protect humanity. Let's use ...
6 votes
0 answers
201 views

### In an infinite maze of squares with p% walls, will you be boxed in?

If you are in an infinite maze of squares, where each border between two squares has a $p$% chance of being a wall, what is the probability $F(p)$ that you are trapped in a finite space? For a better ...
2 votes
1 answer
252 views

### Good and bad numbers of remaining mines

You've been tasked with finishing solving this Minesweeper board: "How many mines remain?", you ask. "I'm just choosing that now, actually. Tell you what: I was going to consider every ...
9 votes
1 answer
410 views

### An island, a boat, and a roulette

You are on an island where a volcano is going to erupt anytime soon, wiping out everything with it. The only way to leave the island is via boat, but the captain of the boat asks for £80 for the ride. ...
7 votes
2 answers
1k views

### The Birthday Paradox, revisited

My niece, a medical consultant, sees a number of patients every day, and the first thing she asks is for their birthday. Recently, in a certain week, patients 16 and 17 shared the same birthday (...
1 vote
1 answer
267 views

### Mastermind algorithm but for unique colors only

I have seen Donald Knuth's Algorithm to guess the correct pattern in maximum 5 tries. However, I want to write a program to find the correct pattern effectively however the code maker can only make a ...