Lesson 2

Three Doors, One Prize

Pick a door, watch one empty door open, and notice why the other closed door gets the bigger chance

The door puzzle

Story: You are on a game show with 3 doors. One door hides the prize. Two doors hide goats. You choose first. Then the host opens one losing door.

At the end, you must decide:

Do the first few rounds slowly. Watch what the host is allowed to open before you decide.

Walk through it slowly

Step 1

You pick 1 door out of 3. At that moment, your door has only a 1 in 3 chance of hiding the prize.

Step 2

The host opens a different door and shows an empty one. The host never opens the prize door on purpose.

Step 3

Your first door still keeps its 1 in 3 chance. The other unopened door now holds the remaining 2 in 3 chance.
1 / 3

Play it yourself

Play 4 slow rounds first. Read the little clue cards each time. Then try a batch of staying rounds and a batch of switching rounds.

Pick a door to start!

Fast mode unlocks after 4 slow rounds.

Big idea: the host is giving you a clue

It can look like 50-50 because only two doors are left. But the host did not open a random door. The host had to open a goat door.

In 10 000 simulated games:

Why switching helps

Your first pick was right only 33.3% of the time. That means it was wrong 66.7% of the time.

When the host opens one empty door, the prize does not magically move. The host is helping you see where that missing chance is hiding: in the other unopened door.

If your first pick was right: staying wins, switching loses.

If your first pick was wrong: switching wins, because the host removed the other wrong door.

So the simple version is:

Because first guesses are wrong 66.7% of the time, switching wins 66.7% of the time.

Quiz time

In the Monty Hall problem, what is the probability of winning if you always SWITCH?

Why does the host opening a goat door give you useful information?

Imagine there were 100 doors. You pick 1, and the host opens 98 goat doors. Should you switch to the last closed door?