The Monty Hall Problem

This is probably the single most argued-about probability puzzle, including among professional mathematicians when it first went viral. Getting the reasoning right, not just the numerical answer, is what an interviewer is actually testing.

The puzzle

There are three doors. Behind one is a car; behind the other two, goats. You pick a door, say door 1. The host, who knows what's behind every door, always opens a different door that has a goat behind it (never the car, and never your door) — say door 3. You're now offered the choice to stick with door 1 or switch to door 2. Should you switch?

Yes — switching wins with probability $2/3$, sticking wins with probability $1/3$.

Why switching wins twice as often

Track what your first pick locks in. When you first choose a door, you have a $1/3$ chance of having picked the car and a $2/3$ chance of having picked a goat — that's just the three doors being equally likely, and it never changes, because the host's later action doesn't alter what was behind your original door.

Case A: you initially picked the car (probability $1/3$). The host opens one of the two remaining goat doors (either one, his choice), and switching takes you to the other goat. Switching loses in this case.

Case B: you initially picked a goat (probability $2/3$). The other goat is behind one specific other door, and the car is behind the remaining door. The host, forced to reveal a goat and not your door, has exactly one legal choice — the other goat door — which he must open. That leaves the car sitting behind the one remaining door. Switching wins in this case.

Combine. Switching wins exactly when case B happened, which is probability $2/3$. Sticking wins exactly when case A happened, probability $1/3$.

Scale the puzzle to 100 doors, one car, host opens 98 goat doors after your pick, and the asymmetry becomes obvious: your first pick is right with probability $1/100$, so the single door the host pointedly leaves closed is right $99/100$ of the time. This conditional-probability trap — mistaking "fewer options remain" for "options are now equally likely" — is exactly the error that misprices information in trading: a market's remaining possibilities aren't equally likely just because some have been ruled out by an informed action.