Questions tagged [probability]
A mathematical puzzle whose essential nature involves randomness. Use with [mathematics]
378
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The Truth And Lies Today: Revisited
A man is only capable of telling the truth four hours a day, and they are either TTFTT or TFTTT (equiprobable). On the other twenty hours, he lies. All hours between 7pm and 6am are night, 7am and ...
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The Truth And Lies Today
Suppose a man only tells the truth three hours a day.
It is the same every week and it is known that the pattern of the truth hours are TFTFT.
(which means he may tell the truth at midnight and 2am ...
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2
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Who is most likely to win all the chips?
Alice, Bob and Charlie each have 5 chips. Starting from Alice, they take turns to act in the order of Alice, Bob and Charlie. The $n$-th person to act must take a total of $n$ chips from their ...
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I'm not a robot, I'm not a MONKEY
The city of Roboville has been invaded by robots, monkeys, and dancers, who will dance if the beat's funky. Some of the invaders can be two of them (e.g. both monkeys and robots) and some can be all ...
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Will Romeo meet Juliet? [duplicate]
Romeo and Juliet try to meet each other everyday at a certain place between 12:00 and 13:00. They arrive at the place at a random time between 12:00 and 13:00 and they wait 15 minutes (but never after ...
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Bertrand's Ballot Theorem
A total of X voting papers for candidate A and Y voting papers for candidate B were cast in one section during the election. Where X > Y
At the end of election day, the voting papers were counted ...
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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 ...
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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 ...
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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 ...
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1
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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 $n$ ...
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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 ...
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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, ...
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2
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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% ...
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2
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2
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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 ...
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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....
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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 ...
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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 ...
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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 ...
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2
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530
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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. ...
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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?
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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 ...
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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 ...
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How should you bet?
Alice and Bob are playing a betting game. Each has 20 chips to start with. They move alternatively starting with Alice. In their moves, they name a number $n$ and toss a fair coin: if head, they get $...
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How many gold coins can you extract from the billionaire?
An eccentric billionaire plays a game with you. She has an urn with 100 gold coins.
Each time, you can take any number of coins from the urn.
If you take n coins, she will flip a fair coin.
If head, ...
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4
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Simulating a biased coin with an unbiased one
You are given an unbiased fair coin, ie., a coin that produces heads with probability 0.5. Can you use this coin to simulate a biased coin that produces heads with some given probability $p$ ?
This is ...
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Simulating an unbiased coin with a biased one
You are given a biased coin. With probability $p$ it produces heads, where $p$ is in the range $(0,1)$. How can you use this coin to simulate an unbiased fair coin, ie., a coin that produces heads ...
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Slow & Fast drivers
Let us define "fastness" as a characteristic of how fast/agressive a driver is. One aspect of this is the (maximum) time between a traffic light turning red and the driver deciding they have ...
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Even odds from die with single mark
We want to make a close equivalent of a fair coin throw, by throwing an N-sided fair die T times. The catch: only one side is marked, by a colored disk on that side's center (like the traditional ...
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Which die should you choose?
You are playing a game with dice. You have a grid with 13 cells shown below. A token starts in the left-most cell (start). Each turn you roll an $n$-sided die and get a number $x$ between $1$ and $n$, ...
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3
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Outmaneuver your opponents in the duel
A, B and C are in a three way duel. Starting with A, they rotate in the order of A-B-C, each firing one shot at a time. They stand close to one another, so that each can kill one of the others or ...
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Blindly choosing a ball from a bag [closed]
You are blindfolded. You are handed a bag and told that there are three balls and two cubes, all of which are the same size and are equally likely to be chosen. You blindly choose one of the items ...
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2
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Balancing the marble game (Red marbles are OP, please nerf)
Ash and Bree are playing a simple game of chance:
They fill a bag with small marbles coloured red or blue.
They take turns to draw a marble from the bag without looking, then:
If Ash draws a red ...
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2
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510
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Lightbulbs in a 3×3 square
Suppose we have a $3\times 3$ arrangement of lightbulbs and we switch them on/off randomly (probability $½$). What is the probability the no adjacent bulbs are on?
My attempt was:
Let $1= $ on and $0 =...
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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 ...
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"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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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).
...
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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: ...
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