Many probability questions are counting questions in disguise. Shared birthdays, best-of-seven series, couples kept apart at a dinner table: the trick is usually a complement, a symmetry or inclusion–exclusion. These puzzles train you to spot which one, and the gut-first format shows how far your instinct is from the count.
What these puzzles practice
Complements: count what doesn't happen
Symmetry arguments
Inclusion–exclusion
19 puzzles, easiest first
No two dice alike· Dice · difficulty 1 of 5You roll 4 fair six-sided dice. What is the probability that they all show different numbers?
Someone shares your birthday· Office party · difficulty 1 of 5You walk into a party with 23 other people. Assume everyone's birthday, yours included, is independent and uniform over 365 days (ignore…
The Chevalier's bet· Dice · difficulty 1 of 5You roll a fair 6-sided die 4 times. What is the probability that at least one roll shows 6?
Ants on a triangle· Brainteaser · difficulty 2 of 5An ant sits on each corner of a regular polygon with 3 corners. At the same moment, each ant picks one of the two edges at its corner,…
Best of three· Sports · difficulty 2 of 5In a playoff series, the stronger team wins any single game with probability 0.6, independently of the other games. The series is best of…
Birthdays in the locker room· Sports · difficulty 2 of 5A team's locker room has 23 people in it. Assume birthdays are independent and uniform over 365 days (ignore leap years). What is the…
Dealt a full house· Poker · difficulty 2 of 5You are dealt 5 cards from a well-shuffled standard 52-card deck. What is the probability that your hand is a full house: three cards of…
Exactly half heads· Coin flips · difficulty 2 of 5You flip a fair coin 20 times. What is the probability that you get exactly as many heads as tails, 10 of each?
Galileo's dice· Dice · difficulty 2 of 5You roll 3 fair six-sided dice and add up the numbers. What is the probability that the total is exactly 10?
Holding an ace· Cards · difficulty 2 of 5You are dealt 13 cards from a well-shuffled standard 52-card deck, which has 4 aces. (A bridge hand is 13 cards.) What is the probability…
Exactly one pair· Poker · difficulty 3 of 5You are dealt 5 cards from a well-shuffled standard 52-card deck. What is the probability that your hand is exactly one pair: two cards of…
Grab a matching pair· Home · difficulty 3 of 5A drawer holds 5 pairs of socks, every pair different, so 10 socks in all. In the dark you grab 4 socks at random, without replacement.…
Never going down· Dice · difficulty 3 of 5You roll a fair 6-sided die 4 times. What is the probability that no roll is smaller than the roll before it? Repeats are allowed, so 2,…
The unlucky second seed· Badminton · difficulty 3 of 5A badminton tournament has 16 players in a single-elimination draw, with the draw made completely at random (no seeding). The better…
Three numbers or better· Lottery · difficulty 3 of 5A lottery draws 6 different numbers at random from 1 to 49. Your ticket has 6 different numbers of your choice. What is the probability…
A streak in twenty flips· Coin flips · difficulty 4 of 5You flip a fair coin 20 times. What is the probability that somewhere in the sequence you get at least 4 heads in a row?
No couple side by side· Wedding dinner · difficulty 4 of 54 married couples (8 people) sit down in a single row of 8 chairs at a wedding dinner, in a uniformly random order: all 8! seatings are…
Nobody draws themselves· Office party · difficulty 4 of 510 friends write their names on slips, put them in a hat, and each draws one slip at random for Secret Santa. What is the probability that…
Ahead all night· Election counting · difficulty 5 of 5In a two-candidate election, A receives 6 votes and B receives 4. The ballots are counted one at a time in a uniformly random order. What…