Questions tagged [probability]

A mathematical puzzle whose essential nature involves randomness.

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9
votes
1answer
516 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: ...
9
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3answers
2k views

Sliding Bolt Puzzle - fastest solution (time-wise)

This is a follow-up question for the Sliding Bolt Puzzle. If you have not solved it yet, you might want to head there first, as the extended discussion of its solution in this question will contain ...
9
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2answers
633 views

What is the probability that all the bullets will be destroyed?

A gun shots bullets with different speed each second. The bullets form a straight line. The bullets keep their speed (not getting slower or faster). But if 2 of the bullets collide, 2 of them are ...
9
votes
2answers
204 views

What are the odds that there is no solution on a given Set board?

I was playing a game of Set with a group. We had the usual twelve cards out and all agreed that there was no set. We then increased the number of cards to fifteen and were still unable to find a set....
9
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2answers
547 views

Generalizing the “shooting puzzle”

Inspired by this puzzle. You have $n$ shooters having a duel, labeled $1, 2, ... n$. Each shooter has an accuracy of $k_n$, with each $k_n$ some probability in $(0,1]$, with $k_1 < k_2 < ... &...
8
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7answers
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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 ...
8
votes
7answers
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 ...
8
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2answers
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. ...
8
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2answers
1k views

Keypad Door Lock

Your progress is barred by a locked door, it is clearly controlled by a keypad. with 10 digits (0-9) plus a button for ENTER and a button for CANCEL. You know several things about this lock: This ...
8
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4answers
976 views

Social Engineering (The Birthrate Problem)

The distant country of Toomalia has a birthrate of 67 boys for every 33 girls. In an effort to restore a 50:50 gender ratio in the country, Toomalian policymakers institute a mandatory new social ...
8
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5answers
2k views

Three 50/50 guesses

Three friends enter a quiz-show as a team, the final round of the show is all or nothing, they only get their prize if they can win this round. Three boxes labelled 1, 2 and 3 are placed in a room, ...
8
votes
2answers
496 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 ...
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 ...
8
votes
3answers
762 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 ...
8
votes
3answers
1k views

A first choice die

You and your friend have met to play a game of dice. This is a very simple game where both players roll a die at the same time, and the player with the lower score pays a coin to the other player. In ...
8
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5answers
413 views

Two person game, choosing an integer between 1 and 30?

Suppose two players are playing a game. The first player chose an integer, n st. $n \in \left\{ 1,\ldots ,30\right\}$. The second player picks a different one. After that, a random integer x st. $x \...
8
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2answers
570 views

Help! I lost my marble(s)!

Tomorrow is the annual meetup for the Northwest Region Marble Lovers Association. The days festivities will include various marble games, admiring collections of marbles, a guest speaker who will talk ...
8
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2answers
314 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 ...
8
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8answers
597 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 ...
7
votes
9answers
7k 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 ...
7
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3answers
1k views

Surviving Captain Nefarious

Captain Nefarious has captured you and two of your friends, Alice and Bob. Being a villain, he has a natural desire to monologue, so he sits you down (guarded, of course by his Largely Incompetent ...
7
votes
1answer
248 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. ...
7
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3answers
2k 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?
7
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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 ...
7
votes
2answers
2k views

Find Coin's unfairness with only 4 tosses

There is a unfair coin and you want to find out its actual probability, to find the exact probability is actually not an easy task, you need to toss it many many times to get the close to actual ...
7
votes
3answers
528 views

Building the ideal Scrabble dice set

This is a genuine question and I don't know the "correct" answer to this. I'm also of two minds if it is really on-topic on the site. Finding the answer is surely "a puzzle" (if possible), but it ...
7
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2answers
728 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 ...
7
votes
1answer
652 views

Will You (Yes, You) Decide The Election? [closed]

You are the only sane voter in a state with two candidates running for Senate. There are N other people in the state, and each of them votes completely randomly! Those voters all act independently and ...
7
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2answers
334 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 ...
7
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4answers
757 views

Born to lie, to live, to die

There is a race which has a truth gene. It consists of 3 alleles, namely T, F and O. A person inherits one allele from each of his parents, hence he/she has always exactly two alleles. For example, if ...
7
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3answers
140 views

Playing Chess Game of Chance

I still enjoy playing chess at the age of ninety four.Those contemporary opponents I used to play with have all passed away without me even getting even scores with any of them. On my spare time I ...
7
votes
1answer
163 views

The Attack on Fort Alpha

I made this puzzle up in 1999 when I was a student. Fort Alpha, on your border territories, was attacked by The Enemy. You don't know if it fell, or repulsed The Enemy's vicious onslaught. A totally ...
7
votes
3answers
435 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 ...
6
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6answers
5k views

100 pieces 1 opportunity, choose wisely!

Say you have a cake that is cut into 100 pieces. One of the pieces has a hidden prize. 100 people take turns, each taking a piece in sequence hoping to find the prize. Note: I slightly modified (...
6
votes
5answers
3k 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 ...
6
votes
5answers
2k views

Ten targets you need to hit

You are a good sniper and there are 10 targets you need to hit. The chance of hitting any target is the same. Strangely, with your accuracy the chance of getting 8 hits is equal to getting 7 hits in ...
6
votes
2answers
454 views

Twin Probabilities Problem

1 in 8 sets of twins are sets of identical twins. While talking to Jessica, you learn that she has a twin named Emily. What's the probability that Emily and Jessica are identical twins? The answer ...
6
votes
2answers
765 views

Drawing balls from a pocket

I have a pocket of N=4 balls of different colors, say green, red, blue and yellow. You can draw as many times as you want After each draw, the ball will be put back to the pocket so that the chance ...
6
votes
2answers
722 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 ...
6
votes
4answers
380 views

Drawing uniformly using a single biased coin

Your task is to draw uniformly from a set of size $N$. You are given a single coin which you can ‘bend’ to control the probability $p\in[0,1]$ of throwing a Head, but you are only allowed a bounded ...
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 ...
6
votes
3answers
373 views

Guessing game with infinity coin flips

Evan and Ollie are going to play a game as a team. The two friends placed in separate rooms, and then a coin is flipped infinitely many times*. Evan is told the results of the even numbered flips, ...
6
votes
2answers
310 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 ...
6
votes
1answer
1k views

cheating on the Russian roulette

This question is inspired by this one. You and your opponent play Russian roulette with two bullets. Quoting from the linked question: You are challenged to a game of Russian roulette. Your ...
6
votes
3answers
524 views

Monty's Labyrinth

(A fun combination of two classic puzzles, hope seasoned puzzlers enjoy.) Having escaped through almost all of a labyrinth, you are met by three magical doors. Two of them lead to certain and ...
6
votes
2answers
585 views

A jar of mixed coins [closed]

This is a slight adaptation of a google interview question that I found entertaining. You are given a jar with N coins, K of them are fair. The remaining coins have heads on both sides. You choose a ...
6
votes
1answer
254 views

Russian Roulette (Redux!)

You are challenged to a game of Russian roulette with a six-chamber revolver, but you and your opponent are allowed to change the outcome of the game in different ways. Here's how the game will ...
6
votes
2answers
608 views

Probability of nobody being born in a specific month

Assume 11 persons are born in 11 different months. What is the probability $P(n)$ that none of them is born in the $n$-th month? This would be pretty easy if all months had the same length, then $P(n)...
6
votes
3answers
351 views

Intersecting bug trail [closed]

A man picks up a huge circular dart board and shoots paintballs at it. $n$ paintballs manage to hit it. They are random and point-sized, for the purpose of this question. A bug is somewhere on the ...
6
votes
1answer
619 views

What is the expected length of the longest piece?

If a rod of unit length is broken into $n$ pieces, what is the expected length of the longest piece? The positions at which the rod is broken are chosen randomly uniformly. This is a generalization ...