#probability

100 prisoners are put in separate cells and taken one at a time, in an order chosen by the warden, to a room with a light switch that is initially off. Every prisoner will be brought to the room again and again, but no one learns anything except the state of the switch. On each visit a prisoner may…

A floor is ruled with parallel lines a distance $d$ apart. A needle of length $\ell \le d$ is dropped at random. What is the probability that it lands across one of the lines?

On a game show there are three doors, one hiding a car and two hiding goats. You pick a door. The host, who knows where the car is, always opens one of the other two doors to show a goat and then offers you the chance to switch to the remaining closed door. Should you switch, and what is your…

100 prisoners numbered 1 to 100 face a room with 100 drawers, into which the numbers 1 to 100 have been placed in random order. One at a time, each prisoner enters, opens at most 50 drawers looking for their own number, and leaves the room exactly as it was, with no communication afterwards. All are…

Assume birthdays are independent and uniformly distributed over 365 days. What is the smallest number of people for which the probability that at least two of them share a birthday exceeds 1/2?

Two players of equal skill play a series of fair rounds, and the first to win 6 rounds takes the whole stake. The game is interrupted when one player leads 5 rounds to 3. How should the stake be divided fairly?

$n$ guests check their hats, and the hats are handed back in a uniformly random order. Let $p_n$ be the probability that no guest gets their own hat. Find a formula for $p_n$ and its limit as $n\to \infty$.