Questions tagged [probability]

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

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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 ...
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2 Monkeys on a computer [closed]

(a) Two monkeys are typing capital letters (A-Z) randomly. The first stops typing when the word COCONUT appears as seven successive letters. The second stops typing when TUNOCOC appears; TUNOCOC is ...
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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 ...
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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 ...
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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 ...
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How many tries to roll a 6?

Suppose you roll a (fair, 6-sided, perfectly ordinary) die repeatedly until you roll a 6. As is well known, the expected (i.e., long-term average over many trials) number of rolls required is 6. Now ...
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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, ...
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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 ...
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Relabeling two 20-sided dice without changing their total

Usually, 20-sided dice are labeled with the numbers 1 to 20. When you roll two of these, their sum is a random number between 2 and 40. The total 21 is most likely to occur, while 2 and 40 are the ...
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Crippled King Crossing a Canyon

A chess king has been injured in battle against an evil wizard, and can no longer move northeast or southwest. This king is on the North rim of a canyon, and must flee to safety on the South rim. ...
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Aproximating 100 by 6

You're a fan of pen & paper RPGs and have finally arranged a new playing group, recruited amongst your friends from the mathematical faculty. As a GameMaster (GM) you've assumed that all your ...
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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 ...
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Do better than chance

You and an opponent are playing the following game: Your opponent picks two different real numbers, however he wants to, and writes them on two cards, which he presents to you face down You choose ...
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Say 100 and win

Our good friends that like games, Alice and Bob, play a game called "Say 100 and win". The rules are: In the beginning they use one regular 6 faced die to get a random number between 1 and 6. Let's ...
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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 ...
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Monty Hall Revisited: Winning Both Goats!

Suppose you're at a game show, knowing the winning strategy for the classic Monty Hall problem. Your turn comes up and you take the stage. You confidently choose door number (?). Just as you are ...
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A Strangely Familiar Probability Problem

You're taking the final exam for your Introduction to Probability and Statistics course, and once again you've failed to study (you were up all night solving online brainteasers). The exam is ...
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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%...
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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 ...
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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 ...
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Shooting Free Throws

Shaq is shooting free throws. He makes his first shot, then misses his second. Confidence is a huge factor in how well Shaq plays. This means that for each subsequent shot, the probability he makes ...
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What is the expected value of this coin flipping game?

A casino offers the following game: You flip a fair coin repeatedly and record the results on slips of paper, one result per slip. At any time you may decide to stop, at which time one of the slips ...
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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 ...
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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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How can the "Boy or Girl" paradox be worded unambiguously?

The first time I encountered the Boy or Girl paradox I was struck by how ambiguous all of the formulations in English are. As Wikipedia notes: Gardner argued that a "failure to specify the ...
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Blindfold Bingo

Congratulations, you have been selected from among the audience to compete in the fabulous Blindfold Bingo challenge! (cue annoying, yet catchy theme music.) Here you see our perfectly ordinary Bingo ...
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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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How to get a uniform 1 out of 7 chance, using only a coin and a 6-sided die [closed]

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 ...
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Contagious ants on a stick

Twenty-five ants are scattered on a meter stick. At the same time, they each pick a random direction (east or west) and start marching at 1cm/second. Whenever two ants meet, they turn around. When an ...
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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 ...
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"Musical chair" for clock hands

A colleague and me usually make bets on the time when a simulation is finished. Next time I would suggest the following: At the time the simulation is finished we take a look at the analog clock on ...
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Two Dice = Nine Cards

I have nine cards, each labeled with a real number (possibly negative or fractional) in a special way. When I randomly pick two of these cards without replacement and add their labels together, the ...
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