Conditional probability is where intuition fails most often: a fraud alert on a rare kind of payment, a card you've been told about, a door the host chose to open. Interviewers ask these because the quick answer and the right answer are usually far apart. Each puzzle asks for your gut estimate first, then you work it out with Bayes' rule or careful counting, and a simulation shows the truth.
What these puzzles practice
Bayes' rule and base rates
Updating on what you're told, and on how you were told it
Counting the cases instead of trusting the story
17 puzzles, easiest first
A six is showing· Board games · difficulty 2 of 5You roll two fair dice behind a screen. A friend peeks and tells you: at least one die shows a 6. What is the probability that the two…
At least one boy· Family · difficulty 2 of 5A family has 2 children, each independently a boy or a girl with equal probability. You learn that at least one of them is a boy. What is…
At least one red card· Playing cards · difficulty 2 of 5You draw 2 cards at random, without replacement, from a well-shuffled standard 52-card deck (26 red, 26 black). A friend looks at them and…
The flagged transaction· Fraud detection · difficulty 2 of 5Of the card payments a bank's model reviews, 5% are fraud. The model flags 95% of fraudulent payments, but it also flags 10% of genuine…
The host knows· Game show · difficulty 2 of 5A game show has 3 closed doors. One hides a car, the others hide goats. You pick door 1. The host, who knows where the car is, opens 1 of…
Three boxes, one gold coin· Antique shop · difficulty 2 of 5A cabinet holds one box with two gold coins, one box with two silver coins, and one box with one gold and one silver coin. You pick a box…
Which die was it?· Board games · difficulty 2 of 5A bag holds two fair dice: an ordinary 6-sided die and a 20-sided die numbered 1 to 20. You pick one of the two at random, with equal…
Which urn am I drawing from?· Urn · difficulty 2 of 5Urn A holds 3 red balls and 1 blue ball. Urn B holds 1 red ball and 3 blue balls. A fair coin flip picks one of the urns and you don't…
An even number of flips· Coin flips · difficulty 3 of 5You flip a coin that lands heads with probability 0.5, independently each time, until the first heads appears. A friend who watched tells…
Flagged twice· Chip factory · difficulty 3 of 5At a chip factory, 2% of chips are defective. An automated inspection flags 90% of defective chips and 5% of good chips. Each time the…
Is it raining in Seattle?· Travel · difficulty 3 of 5You're flying to Seattle, where the chance of rain on any given day is 25%. You call three friends who live there and ask each,…
Looking back at the first ball· Urn · difficulty 3 of 5An urn holds 3 red and 2 blue balls. You draw two balls, one after the other, at random without replacement, and set them aside without…
The two-headed coin· Magic shop · difficulty 3 of 5A magic shop's bag holds 4 fair coins and one coin with heads on both sides. You pull out a coin at random without looking at it, flip it…
A boy born on a Tuesday· Family · difficulty 4 of 5A family has two children. Each child, independently, is a boy or a girl with probability 1/2 each, and each child is equally likely to…
Ten heads from a mystery coin· Coin flips · difficulty 4 of 5A coin comes out of a machine with an unknown probability of heads, equally likely to be any value between 0 and 1 (a uniform prior). It…
The rich get richer· Lucky draw · difficulty 4 of 5A box holds 2 red and 3 blue balls. Each round you draw one ball uniformly at random, note its color, and put it back together with one…
Which ace did they see?· Poker · difficulty 4 of 5You are dealt 3 cards from a well-shuffled standard 52-card deck. A friend looks at your hand and tells you truthfully: “You have the ace…