Probability

Three points are chosen uniformly at random on a circle. What is the probability all three lie within some semicircle?

Each minute an amoeba dies, stays the same, splits in two, or splits in three, each with probability 1/4. What is the probability the population dies out?

An ant walks randomly along the edges of a cube. How many moves does it expect to need to reach the opposite vertex?

A random chord is drawn by picking two points on a circle. What is the probability it is longer than a side of the inscribed equilateral triangle?

Drop a needle onto a floor ruled with lines exactly one needle-length apart. What is the probability it crosses a line?

There are 6 distinct coupons, each equally likely in a cereal box. How many boxes do you expect to buy to collect all 6?

What is the probability of rolling at least one six in four rolls of a fair die, the bet that helped launch probability theory?

A fair die is rolled six times. How many distinct faces do you expect to see?

Alice and Bob alternate flipping a fair coin, Alice first; whoever flips heads first wins. What is the probability Alice wins?

What is the probability a random 5-card poker hand is a full house?

A gambler starts with $10 and bets $1 on a fair coin flip each round, stopping at $0 or $20. What is the probability of reaching $20?

Roll two fair dice. What is the expected value of the larger of the two values?

You pick 1 of 4 doors. The host opens 2 losing doors. What is the probability of winning if you switch?

A fair coin is flipped 10 times. What is the probability no two heads land consecutively?

Roll a die up to three times; after each roll, keep the value or roll again. What is the expected payout under optimal play?

Draw a ball, return it with another of the same color, repeat. What is the probability the third draw is red?

A symmetric random walk on the integers starts at 0. What is the expected number of steps to return to 0?

Roll a fair die until a six appears, then stop. What is the expected sum of all your rolls?

Two dice are rolled repeatedly. What is the probability a sum of 7 shows up before a sum of 8?

The chance of seeing a shooting star in an hour is 0.84. What is the chance of seeing one in 10 minutes?

A stick is broken at two uniformly random points. What's the probability the three pieces form a triangle?

100 passengers, 100 assigned seats, and one lost ticket. What is the probability the last passenger ends up in their own seat?

Two friends each arrive at a random time between noon and 1 pm and wait 15 minutes. What is the probability they meet?

You interview n candidates one by one and must decide immediately after each interview. What is the probability the optimal stopping rule selects the best candidate?

Two socks drawn at random from a drawer of red and black socks are both red with probability exactly 1/2. What is the smallest possible drawer?

A family has two children and you learn at least one is a boy. What is the probability both are boys?

Flip a fair coin repeatedly. How many flips do you expect to need before seeing two heads in a row?