Problems

Three doors, one prize. Should you switch?

How few people do you need in a room before two share a birthday?

A 99% accurate test says you have a rare disease. How worried should you be?

You and a friend each pick a sequence of three coin flips. Yours is HHH. Whose pattern shows up first?

Hire one person from 100 candidates, one at a time, no callbacks. What's the optimal strategy?

Find the highest floor an egg survives — but you only have two.

Half your batteries are dead. How few trials to guarantee a working flashlight?

One egg is heavier. With only a balance scale, how few weighings do you need?

100 passengers board a plane. The first one is drunk and sits randomly. What's the probability the last passenger gets their assigned seat?

Every family keeps having children until they have a son, then stops. What happens to the boy-to-girl ratio?

Ants walk along a meter-long stick. When two meet, they reverse. How long until they all fall off?

Cereal boxes hide one of n coupons at random. How many boxes do you expect to buy before collecting them all?

1000 bottles, exactly one is poisoned. You have 24 hours and a few prisoners. How few?

Each prisoner can open 50 of 100 boxes. They must all find their own number. Can they survive?

I have two children. At least one is a boy. What's the probability both are boys?

Snap a stick at two random points. What's the probability the three pieces form a triangle?

Flip a coin until you see HH. Or HT. Same expected time, right?

Five rational pirates divide 100 gold coins. What does the most senior pirate keep?

A casino offers a game with infinite expected value. How much would you pay?

A random walker takes one step at a time in a uniformly random direction. Do they always come home?

An ant walks randomly along the edges of a cube. How many steps to the opposite corner?

Your only coin is biased and you don't know by how much. Can you still simulate a perfect 50/50?

A wins a 7-3 election. Votes are counted in random order. What's the probability A leads the entire count?

Pick three uniformly random points on a circle. What's the probability the triangle they form is acute?

Two consecutive bullets in a six-shooter. After an empty click, should you spin before your turn?

N people check their hats. The clerk returns them at random. What's the probability nobody gets their own?

Four people need to cross a bridge in 17 minutes. Only two can cross at once, and they need the only flashlight.

Two ropes each burn for an hour, but unevenly. Measure exactly 45 minutes.

Two trains approach each other. A fly bounces between them until they collide. How far does the fly travel?

100 coins on a table, 10 are heads up. You're blindfolded. Split them into two piles with equal heads.

Five couples at a party. The host asks nine guests how many hands they shook and gets nine distinct numbers.

Tom has the same number of brothers as sisters. His sister Jane has twice as many brothers as sisters. How many children?

Three switches outside a sealed room, one controls the bulb. You may enter only once. Identify it.

Albert knows the month, Bernard knows the day. From their statements alone, find Cheryl's birthday.

3000 bananas, a camel that eats one per mile, and 1000 miles to the market. How many arrive?

100 students toggle 100 lockers at intervals. Which lockers are open at the end?

In any group of six people, there must be three who are all mutual friends or three who are all mutual strangers.

Break an n×m chocolate bar into individual squares. What's the minimum number of breaks?

A coin lands heads 60% of the time. You can bet any fraction of your bankroll. What fraction maximizes long-term growth?

A fair coin, a small bankroll, and a deep-pocketed casino. What's your probability of breaking the house?

Two losing games. Alternated correctly, they become a winning strategy.

A stock goes up 50% then down 50%. Average return is zero. What's your actual return?

You have $12,000 to invest. Buy all today, or drip $1,000 per month for a year. Which usually wins?

Can moving one person between two groups raise the average of both groups simultaneously?

Hospital A has a higher overall survival rate. Hospital B is better at treating *both* mild and severe cases. Which hospital should you choose?

Pick a number 0-100. Whoever's closest to two-thirds of the average wins. What do you pick?

Two pairs of gambles. The choice most people make is mathematically inconsistent — and reveals something deep about utility.

A stream of items of unknown length. Pick one uniformly at random — using one slot of memory.

Captured tanks have serial numbers 14, 37, 89, 156, 209. How many tanks does Germany have?

Drop a needle on a floor of parallel lines. The probability it crosses a line involves π — and π appears in a problem with no circles.