The Monty Hall problem

I’ve long wanted a simple explanation of the Monty Hall problem and I’ve never found one that I liked. Some I really detested like one that tried to make some lame analogy to baseball pitchers.

Anyway, here is what I’ve found to be the simplest explanation yet. First, what’s the problem.

In a game show, the contestant is shown into a room with three identical closed doors. He is informed that behind one door is a prize and behind the other two doors, there is nothing.

He is then asked to pick a door. Once he has picked a door, the host proceeds to open one of the other two doors (that he had not picked) and shows the contestant that there is nothing behind that door.

The host then offers the contestant the option of either changing his selection (picking the third remaining door), or sticking with his initial choice.

What should the contestant do?

The simplistic answer is that once the contestant has been shown that there is nothing behind one door, the problem reduces to two doors and therefore the odds are 50-50 and the contestant has no motivation to switch.

In reality, this is not the case, and the contestant would be wise to switch. Here is why.

image1Three doors, behind one of them is the prize, behind the other two, there is nothing.

The contestant now picks a door. For the purposes of this illustration, let’s assume that the contestant picks the door in the middle as shown below.

image2Since the prize is behind one of the three doors, the odds that the prize is behind the door that the contestant has picked is 1/3. By extension therefore the probability that it is behind one of the other two doors is 2/3 (1/3 for each of the doors).

So far, we’re all likely on solid footing, so let’s now bring in the twist. The game show host can always find a door behind which there is nothing. And as shown below, he does.

image3The game show host has picked the third door and there’s nothing there.

However, nothing has changed the fact that the probability that the prize was behind the door that the contestant chose is 1/3 and the probability that it is behind one of the other two doors is 2/3. What has changed is that the host has revealed that it is not behind the door at the far right. If then the probability that it is behind the far left door and the far right door (the two doors that the contestant did not pick) is 2/3, we can say that the probability that it is behind the far left door has to be 2/3.

With this new information therefore, the contestant would be wise to switch his choice.

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