11 problems
Probability interview questions
Conditional probability, symmetry arguments and distributions. The largest single category in quant interviews and the one candidates most often get wrong under time pressure.
The final roll
HardYou roll a fair six-sided die until the running total first exceeds 12. What is the probability the final roll is a 6?
Two children, one is a boy
Warm-upA family has two children. You are told at least one of them is a boy. What is the probability both are boys?
The three doors
Warm-upThree doors hide one prize. You pick a door. The host, who knows where the prize is, opens a different door and reveals it is empty, then offers you the swap. Do you swap?
First head wins
StandardYou and I alternate flipping a fair coin, you first. The first person to flip heads wins. What is your probability of winning?
Breaking a stick into a triangle
HardA stick of length one is broken at two points chosen uniformly and independently. What is the probability the three pieces form a triangle?
Shared birthdays
StandardHow many people do you need in a room before the probability that two share a birthday exceeds one half?
Waiting for two heads in a row
HardWhat is the expected number of fair coin flips until you see two heads in a row?
Returning to the origin
HardA symmetric random walk on the integers starts at zero. Does it return to zero with probability one, and what is the expected time to return?
The two envelopes
StandardTwo envelopes contain money, one twice the other. You open one and find 100. Swapping appears to have expected value 125. What is wrong?
The test that is usually wrong
StandardA condition affects one person in a thousand. A test detects it 99 percent of the time and has a 5 percent false positive rate. You test positive. What is the probability you have it?
Hats returned at random
StandardN people leave hats at a cloakroom and receive them back in a uniformly random order. What is the expected number of people who get their own hat?