the Monty Hall problem
/ MON-tee hawl /
On a game show there are three doors. Behind one is a car, behind the other two are goats. You pick a door — say door 1. The host, who knows where the car is, opens a different door — say door 3 — to reveal a goat, then offers you the chance to switch to the remaining door, door 2. Should you switch? The famous, counterintuitive answer is yes: switching wins with probability 2/3, while staying wins only 1/3.
Here is the careful reasoning. Your first pick is right with probability 1/3 and wrong with probability 2/3 — that does not change. The key is that the host's action is not random: he always opens a goat door and never the one you chose. So when your original pick was wrong (probability 2/3), the host is forced to reveal the only other goat, which means the remaining door must hide the car. Switching therefore wins in exactly the 2/3 of cases where you were originally wrong, and loses only in the 1/3 where you were originally right. Conditioning correctly on the host's informative choice — restricting to the worlds consistent with 'host opened a goat door' — gives 2/3 for switching.
The puzzle is a perfect illustration of how conditioning on information that someone deliberately revealed differs from conditioning on a random event. The trap is to think the two unopened doors are now symmetric and so 50-50; they are not, because the host's behaviour leaks information about your original door without leaking it about the other. If instead the host opened a door at random (sometimes revealing the car), the analysis changes and the symmetry can return — which is exactly why stating the host's rule precisely is essential.
Enumerate with the car behind door 1 and you picking door 1 (you'll stay vs switch). Across all car placements: if you always switch you win whenever your first pick was a goat — 2 of the 3 equally likely placements — so switching wins 2/3 of the time. Staying wins only when you first picked the car: 1/3.
Switching wins exactly when your first pick was wrong, which is 2 times out of 3.
The 2/3 answer depends on the host's rule: he knows the car's location, always opens a goat, and never opens your door. Change that rule and the answer changes — the puzzle is really about modelling the host correctly.