You participate in a game show. In the final contest the host presents you with three doors. Behind one door is one million dollars. The other two hide sacks of potatoes.
The host asks you to choose one door. After you have done so, the host will open one of the other doors, one hiding a sacks of potatoes.
Now there are only two doors left - the one you’ve chosen, and one more. The host asks you:
“Do you want to switch doors?”
What should you do to maximise your chances of getting the million dollars?
Answer
You should switch doors. This can be tricky to see, but the simple thought experiment below will prove it.
The million dollars can either be behind the left, center, or the right door. Without loss of generality, lets assume it is behind the left door.
First, say that you never switch doors. Then you will get the million dollars if you pick the left door, and a sack of potatoes if you pick the center or right doors. You have a one in three chance of getting the million dollars.
Second, say that you always switch doors. If you pick the left door, you will switch away from the million dollars and get a sack of potatoes. If you pick the center door, the host will open the right door - the host always opens a door hiding a sack of potatoes - and you will switch to the million dollars. The same happens if you pick the right door. This gives you a two in three chance of getting the million dollars.