Of cars and goats

Some days ago, while listening to an exam exercise, I learned about an old story that happened in the early nineties. It seems it stirred quite a controversy at the time. It was about a TV show where a contestant was offered to chose one door out of three to win a car. The setup was that only behind one of the doors a car was present while the two others contained a goat. Apparently, most people value the car as the good prize but could not care less about the goat, that was the losing item.


The host would open a door different than the one the contestant chose to reveal, invariably a goat. And later the host will offer the contestant to switch to the other remaining door. The controversy was about what was the best choice the contestant could do: to switch or to keep the original choice.

We all agree that, initially, the contestant has one of out of three chances to win the car. However, when the host opens one of the remaining (non chosen) doors showing a goat it is not so clear what the chances are. Many people would think that given the fact that only two doors remain closed and one of them contains, necessarily, a car, the contestant has a 50/50 chance to win now. If that were true, switching or staying put would make no difference.

However, the recommendation by Marilyn vos Savant in the article linked above was to "always switch" as a more beneficial policy, which many people failed to understand.

So after my exam, I decided to read the above column and to figure that by myself. The logic of this problem works is as follows: Let us assume contestant has chosen the left door.

This selection creates a divide between the chosen door and the other two remaining doors. It is easy to see that the chances of the car to be in the chosen door is 1/3 and the chances of the car to be on the other remaining two doors is 2/3 (the picture shows one of these latter cases).

And what are the implications of the contestant keeping his original choice? Well, he will win in 1/3 of the cases, most people will easily agree on that too.

So the last question is, what are the implications of always switching once the host has shown us a door with a goat? Well, we are then selecting the box in the right of the image, the one that has a 2/3 choice of having the car, that we know from a fact the car is not on the open door so it will be on the closed door with a choice of 2/3.

Therefore, as odd as it might seem, changing to the other remaining door when the host shows us one door containing a goat is the best course of action to maximize our chances to win. That said, we might lose the game if our initial door choice was right, but that will only happen 1/3 of the times. So always switching will grant us a 2/3 chance of winning.

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