The Magic of Logic (16)
Who hasn’t gone through the Monte Hall problem? So, why not confront Enders and Hennes with it?
Recently, they got a vivid description of the famous probability puzzle: you are in a game show, and the host shows you three doors. Behind one of them is a beautiful new car, behind the other two are sacks of potatoes (in the original version they are goats).
The host tells you that you can choose one door and will get whatever is behind it. But before you open it, he opens one of the other two doors, one that has potatoes. And now he gives you a choice: stick with your original door, or switch to the other unopened door?
“But that makes absolutely no difference,” the boys said, although Enders was suspicious: “It probably does, otherwise you would not have asked us!” Right, that is one way to solve it. But I insisted the boys find a real reason why it might be better to switch your choice.
They were stuck. Two hours of playground adventure later they still didn’t see why one strategy should give you better odds at getting the car than the other. It’s absolutely the same — one in three chances of getting the car.
You may want to think about the problem — I will be revealing the solution below. If you find it mystifying you are in good company: when it was first proposed, a majority of people (87%, many PhDs) insisted that each door has the same probability, and conclude that switching does not change anything.
Here’s a little help the boys got: let us assume there are ten doors, and you choose one. Now the host opens eight of the remaining doors, which all have potatoes behind them, and says that you can switch your choice to the ninth one if you want. “Yes, it is absolutely clear, you must switch!” exclaimed Enders. And now he was able to explain the original three-door Monte Hall problem quite succinctly:
“With three doors, if I choose one door and stick to it, I have a one in three chance of getting the car. If the host shows me one of the doors that has potatoes behind it, then I have a one in two chances of getting it right if I switch! In three games I lose one and win two.”
Correct, Enders! Most people assume that the odds do not change, but by opening a door with potatoes behind it, the host has provided additional information. We worked out exactly why it is better to switch choices:
The car is hidden behind one door. You have a one in three chances of getting it right with your first choice. In that case, if you switch, you get potatoes. If you got it wrong the first time, it means you switch from the bad choice to the car, and that happens two out of three times.
So in one out of three tries it was bad to switch, and in two out of three it was good. There is a 2:3 chance of getting the car if you switch, a 1:3 chance if you don’t.