ODDLY LIKELY
Game

Three doors: switch or stay?

One door hides a car. Pick a door, the host opens a losing one, and then you decide. Play it yourself, then let the computer play it a thousand times.

Pick a door.

When you stayed
0 / 0—
When you switched
0 / 0—

Let the computer play

A few games prove nothing — that's why the puzzle fools people. Run a thousand.

Stay—
Switch—

Why switching wins

When you first pick, you have a 1 in 3 chance of being right. That means there's a 2 in 3 chance the car is behind one of the other two doors.

The host then opens a door — but never the car, and never your door. That 2 in 3 doesn't disappear and it doesn't spread out. It all lands on the single door he leaves closed. So switching wins two times out of three, and staying wins one time in three.

All the cases

Your first pickHow oftenStaySwitch
The car1/3WinLose
Goat A1/3LoseWin
Goat B1/3LoseWin
The part that does the work: the host knows where the car is. If he opened a door at random and it happened to be a goat, switching would gain you nothing. His knowledge is what moves the odds.

Try it with cups

Three cups and a coin. A friend hides the coin, you point at a cup, they lift an empty one they know is empty, and you switch every time. Ten rounds is usually enough to feel it.

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