# Questions tagged [probability]

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

423 questions
Filter by
Sorted by
Tagged with
1 vote
2 answers
238 views

### What's the correct intuition behind the expected number of die rolls until a 6 if all previous are even? [duplicate]

I came across a puzzle on YouTube recently (spoiler). You roll a fair dice until you get a 6. What is the expected number of rolls, including the roll of 6, conditioned on the event that all previous ...
• 23.5k
2 votes
2 answers
405 views

### Choose the wine

I based this on a problem from a mathematics presentation, adding a small twist. I did not readily find it here. Your friend comes to dinner and you know he loves to drink Beaujolais. You have 'Cote ...
2 votes
1 answer
172 views

### Hitting twice with different choices

This game of two players has public parameters an integer $n\ge2$, and a probability $p$ with $1/n<p\le1$. E.g. $n=4$, $p=1/3$. In the first phase of a game, a player secretly decides $n$ ...
• 383
2 votes
1 answer
1k views

### Probability that there will be no mutual best friendships? [closed]

Here is a problem: There are two groups of n users, 'A' and 'B'. Each user in A is friends with those in B, and vice versa. Each user in A will randomly choose a user in B as their best friend and ...
12 votes
2 answers
1k views

### Can a fair coin be used to simulate an arbitrary weighted coin, with on average only two flips? [duplicate]

This puzzle was told to me by a Stanford PhD student in a graduate course on randomized algorithms. It's really clever/beautiful, and I think makes for a great challenge. You are given a fair coin, ...
• 195
4 votes
2 answers
380 views

### A simple guessing game

You and your friend play a simple guessing game. Your friend thinks of a secret whole number between 1 and 10, inclusive. If you ask your friend what the number is, the following happens: With 80% ...
2 votes
2 answers
299 views

### Gambling game where you pick how many coins to flip

At a gambling joint, you are invited to be a player in a game with identical unfair but consistent coins flipped simultaneously in each round, with no cost to enter. Each player gets a different ...
9 votes
1 answer
380 views

### Probability that D wins this contest

Problem: $A, B, C, D$ are playing an archery game. They have equal probability of landing their arrow on any spot the target with radius of $R$. Their score is the distance to the center of the target....
1 vote
2 answers
444 views

### A certain 'Monty Hall' variant was played. How many doors were there, and, what was the value of the prize?

Recall the standard Monty Hall scenario where a presenter hides one prize with high value $h$ behind one closed door (e.g., in the classical version: the one high prize is a car), and, equal prizes ...
16 votes
6 answers
3k views

### Three horse race

There are three horses in the race. You know the following information about them: Horse A will finish the race in 50 or 60 seconds with both events being equally likely. Horse B will always finish ...
0 votes
2 answers
165 views

### How can we allocate when we have 150 open slots every day (5 days a week) for those 200 arrivals every day

My question is to solve a very basic problem related to the allocation of slots. Say there are 20 teams with 10 persons in each team. I have 150 open slots every day (5 days a week) for those 20 teams ...
• 101
4 votes
2 answers
603 views

### How many seats around the table?

This puzzle is very much inspired by a recent, excellent, one, posted here. If anyone, in particular the author of the other puzzle, prefers this one to disappear, I will immediately do so and I ...
3 votes
1 answer
146 views

### Two pairs of dice with the same probability [duplicate]

Alice has two dice each numbered from 1 to 6. Bob has two dice as well, the numbering is not known, but each face has a positive integer. The numbering on each dice can be different and a number can ...
• 12.2k
5 votes
2 answers
911 views

### The Game of MIM

In the game of MIM, each of two players, Alice and Bob, receive a board as shown below: They are then dealt in turns one card from a shuffled set of cards numbered 0 through 9 which they must ...
26 votes
4 answers
4k views

### The "Loop of rope" dilemma

It's a "dilemma" that I encountered in real life problems. I believe that it's not new and is quite easy to explain, but may look puzzling at first glance. Alice and Bob are playing with a ...
• 7,387
3 votes
1 answer
222 views

### What is the most STAMINA-efficient strategy to escape the well?

You are roleplaying as an adventurer under the direction of a sadistic DM who has just thrown your character down a well. In order to escape, you have unlimited chances at a STAMINA check, difficulty ...
• 2,379
2 votes
2 answers
270 views

### Extension to the coin flipping puzzle

This is an extension to this great puzzle: Coin Flipping Puzzle. The main difference is we now have ternary sequences instead of binary sequences. Two criminals $A$ and $B$, were recently captured and ...
1 vote
1 answer
253 views

### Optimal parking at a concert

You are going to a concert. The road to the concert has 100 car parking spaces arranged in a straight line. The spaces are numbered from 1 to 100, where space 1 is in the beginning of your journey and ...
6 votes
1 answer
1k views

### Three cups and a ball

I am walking down the streets of Paris when a street performer approaches me. He has a row of 3 cups facing down on a table. He places a ball under the left cup. He proceeds to swap pairs of cups, ...
10 votes
2 answers
554 views

### How should you distribute the fighting power?

Alice and Bob both have N warriors under their command, numbered 1~N, and $1$ point of fighting power at their disposal. Before the game, they privately distribute the power between their warriors. ...
• 6,498
0 votes
2 answers
177 views

### Odds of there being two solutions to a word puzzle [closed]

What are the odds of there being two solutions to, say, a cryptoquote? And is anyone aware of an example of such?
• 301
6 votes
2 answers
504 views

### A three-way bet

Alice, Bob and Carole are playing a betting game. At the beginning, they have 20 chips in the common pot. They move alternatively in the order of Alice, Bob and Carole. In their move, a player ...
• 6,498
5 votes
3 answers
768 views

### Number of cards to be pulled

Ten cards, numbered 1-10, are randomly dealt to 2 players, A and B, such that each player gets 5 cards. Before the cards are dealt, player A will say a number N. After the cards are dealt, A will ...
• 874
12 votes
1 answer
462 views

• 61
1 vote
1 answer
253 views

### Variation of 100 prisoners [duplicate]

There are 100 prisoners, each prisoner is given a number between 1 and 100 without knowing which number, the number can repeat and not all of the numbers must be given. Each prisoner in his turn can ...
• 11
7 votes
9 answers
3k views

### "We accept 20% of candidates and I'm 90% good at my job..." [closed]

There is a famous investment firm where a partner was known for having candidates walk into the room without a greeting, sit across from him at a huge desk, and start right away by asking: "We ...
• 211
1 vote
2 answers
597 views

### Anne and Bernard's marbles

Anne and Bernard have between them 100 marbles, Anne the most. Some in each bag of their marbles are red, the rest, though fewer, are blue. If Anne draws at random two marbles from her bag of marbles, ...
7 votes
1 answer
296 views

### Stochastic Taxicab Path

A city's roadworks is laid out as a perfectly rectangular tiling. A commuter within this city has to travel to work a distance of 17 blocks east and 7 blocks north each day, and tries to take the same ...
• 2,379
0 votes
2 answers
111 views

### Car registration similarity [closed]

In my city, car registration plates contain 3 numbers (0 to 9) and 3 letters (A to Z). Today I've noticed that my neighbour's car has the same registration as my car except for one character. Should I ...
10 votes
3 answers
751 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 ...
9 votes
5 answers
552 views

### The Two Envelope Game Show

This is an original riddle I made, heavily inspired by the Two Envelope problem. I have my answer to this problem, but (full disclosure) even I’m not 100% sure I’ve got it right. I post it half ...
• 410
19 votes
1 answer
898 views

### Honeybee Hangover

A drunken honeybee lands on a completely random hexagon of a large triangular section (depicted below) of its hive, and then every second afterwards, takes a step to a completely random adjacent ...
• 2,379
18 votes
1 answer
653 views

### Lap Theory Optimal Betting Game

Three prisoners are seated at a table. Each of them has a mobile phone on their lap, and they are not allowed to look at anyone else’s phone (and obviously no other form of communication is allowed). ...
• 7,594
9 votes
2 answers
560 views

### Three way duel, which guns to choose?

Alice, Bob and Carole are involved in a game of three way duel. They take turns to shoot in the order of Alice-Bob-Carole-Alice-... until only one surviver is left standing. The rule is very simple: ...
• 6,498
11 votes
1 answer
888 views

### Lawbreaking by the numbers

Special Agent Benford has been stymied by a third unsolved case of miscreant creative accounting and will voluntarily retire from the Fraudulent Numbers Task Force. Benford’s ability to detect ...
• 21.9k
11 votes
5 answers
1k views

### Two player Monty Hall variation

Alice and Bob are going on a game show as partners. In this game, there are three doors, which contain a car, the car keys, and a goat, which are arranged randomly and secretly behind the doors. Alice ...
• 4,704
6 votes
4 answers
1k views

• 5,600
1 vote
3 answers
219 views

### Random sum adds to 1000 [closed]

A random whole number is generated between 1 and 1000 inclusive, and added to a sum. A new randomly generated whole number is generated and added to the sum until the total sum is greater than or ...
• 1,025
6 votes
5 answers
1k views

### Probability of three identical digits on a four-digit digital clock?

On a digital clock showing 00:00 to 23:59, what is the probability of having exactly three identical digits?
• 12.2k