The Monty Hall Trap

Now the offer: stick with your door, or switch to the last unopened one. Most people feel it’s a coin flip — two doors left, 50/50, why bother moving? Hold onto that feeling. We’re about to test it.

You may have seen this exact puzzle before. In the movie 21, an MIT professor springs it on a student to test whether he’ll think or just guess. Watch how fast intuition and probability part ways:

Don’t take anyone’s word for it — play a few rounds. Pick a door, see the host reveal a goat, then choose. Keep an eye on the two tallies at the bottom.

Before the math: if you played a thousand games always switching, what win rate would you expect? Commit to a number in your head.

The more trials you run, the tighter the rates lock onto $\tfrac{1}{3}$ and $\tfrac{2}{3}$. So where does the missing intuition go? Why isn’t it a coin flip?

Your first pick was right $\tfrac{1}{3}$ of the time and wrong $\tfrac{2}{3}$ of the time. The host’s reveal doesn’t change that — but it quietly sweeps the entire $\tfrac{2}{3}$ of “you were wrong” onto the single remaining door. Look at all three equally likely cases:

Switching turns every “you first grabbed a goat” into a win — and that happens twice as often as grabbing the car. The host hands you the $\tfrac{2}{3}$ he was holding.

Let’s put numbers on the gut feeling. When you first point at a door, you’re right with probability $\tfrac{1}{3}$ and wrong with probability $\tfrac{2}{3}$. That split is fixed the instant you choose — before the host touches anything.

Now the “always switch” rule: switching wins exactly when your first pick was a goat. That happens $\tfrac{2}{3}$ of the time. So switching wins $\tfrac{2}{3}$ and staying wins $\tfrac{1}{3}$— they’re just the two outcomes of that very first guess, seen from opposite sides.