Warm-up before harder conditional probability.
The three doors
The question
Three 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?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
Yes. Swapping wins 2/3 of the time.
Your first pick is right 1/3 of the time and wrong 2/3 of the time. The host's action does not change that, because the host can always open an empty door whatever you picked.
In the 2/3 of cases where your first pick was wrong, the prize sits behind the one door the host did not open, so swapping wins.
In the 1/3 where you were right, swapping loses. Swapping therefore wins with probability 2/3.
Extend it to a hundred doors if the intuition resists: you pick one, the host opens ninety eight empty ones, and swapping now obviously wins ninety nine times in a hundred.
What it is testing
Whether you can separate new information from an action that carries none.
Try these next
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?
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?
Preparing properly
Live opportunities
Interviewing soon? See who is hiring
Open vacancies we track at quant firms, grouped by role and city. Adverts are anonymised.
Hiring for these roles? Brief the desk on a hire or check your fit for a role.
Research desk
Speak to a quant market specialist
Ask what a firm really pays, how a desk hires, or where your profile sits. One reply from the research desk, in confidence, no obligation.