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20. A Cadillac or a Goat?

The Monty Hall Paradox

The Question:

The television show Let’s Make a Deal, hosted by the game show host Monty Hall, was very popular in American in the 1960s. On the stage there were three doors. Behind one of the doors was a Cadillac, behind the other two there were goats. If the contestant chose the correct door, the Cadillac would be his.

But there was an additional twist. Right after the contestant indicates his choice, but before the door is opened, Monty Hall, the TV host, opens one of the other two doors to reveal a goat. Then he gives the contestant a chance to revise his choice of doors: do you want to stick with your original door or would you like to switch to the other still unopened door?

Since there are now only two doors left, the probability of the Cadillac being behind either one, is ½, correct? Hence, it should not make any difference whether the contestant remains with the original door or switches to the other one, right?

 

The Paradox:

Nope. Totally wrong!

To increase the chances of winning the Cadillac, the contestant should switch doors.

But how can that be? There are two doors left and the Cadillac could be behind either one of them. So the probability must be one half each, no? The fact that Monty Hall revealed a goat hidden behind one of the other doors is quite irrelevant, is it not? The shenanigans with the third door seem suspiciously reminiscent of the ‘Independence of Irrelevant Alternatives’ (see Chapter ///) and should be ignored, no?

No, no and no.

 

Background:

As it turns out, if the contestant switches doors, the chances of winning the Cadillac are doubled. Even though two unopened doors remain, the winning probabilities are not ½ each. In fact, the chances of winning the Cadillac are only 1/3 if the contestant remains with the original door and surprise, surprise, switching to the other door increases the probability of winning the Cadillac to 2/3. We’ll see below why this is so.

The paradoxical situation was raised again in the 1990s in a series of magazine articles by Marilyn vos Savant, a famously cerebral columnist (according to the Guinness Book of Records she had the world’s highest IQ). She gave the correct answer – one must switch doors – whereupon she received thousands of critical letters, many of them by readers with PhD’s, and many of them calling her an idiot.

The angry readers may be forgiven for their error, if not for the vile language, because the correct answer is indeed quite counter-inuitive. It even threw off one of the most famous mathematicians of the time, the number theorist and probabilist Paul Erdös. He could only be convinced, and even then only reluctantly, after a friend performed simulations on his computer, and showed him that by changing the doors the winning probability increased to 2/3. [1]

 

Dénouement:

At the outset, the probability that the Cadillac is hidden behind any specific door is 1/3. Hence, when the contestant makes a choice, say for door number 1, the chance that he or she wins the Cadillac is 1/3. On the other hand, the chance that he or she goes empty-handed – because the Cadillac is hidden behind one of the two other doors – is 2/3.

Now let’s say that the host reveals a goat behind door number 2 and the contestant remains with the choice of door number 1. The probability that he or she wins the Cadillac does not change just because the host opened some other door. It is still 1/3. What does happen, however, is that the entire remaining probability of 2/3 is now collapsed to door number 3. Hence, there’s a 1/3 probability of the Cadillac being behind door number 1, and a probability of 2/3 that the Cadillac is behind door number 3. The contestant would do well to switch from door 1 to door 3!

The correctness of the switching strategy can also be seen in the following manner: with their first pick, contestants, unbeknownst to them, would be lucky 1/3 of the times. On this 1/3 of all trials, sticking with the door always wins, switching always loses. In the other 2/3 of the trials – when contestants, unbeknownst to them, choose a door with a goat – Monty Hall opens the other goat door. On these 2/3 of all trials, switching always wins and staying always loses. So, remaining with the chosen door wins 1/3 of the times, switching wins 2/3 of the times.

 

Technical supplement:

There are several possible explanations why people believe that the chances of the Cadillac being behind each of the remaining doors are ½ each. One reason is that people strongly tend to believe that probability is evenly distributed across all possible alternatives, whether it is true or not.

Other explanations are behavioral: people tend to overvalue the winning probability of the already chosen door since they already ‘own’ it. And people also may regret errors of commission more than errors of omission: if they lose the Cadillac because they explicitly decided to switch the door, they would kick themselves more than if ‘fate’ had just chosen the other door. Hence, thinking ahead about their possible future regret, they prefer to stick with the choice of door they have already made.

* * *

In an experiment, published in 2010, under the title ‘Are Birds Smarter Than Mathematicians?’, the two scientists Walter T. Herbranson and Julia Schroeder performed Monty Hall trials with pigeons. The birds were confronted with three response keys and mixed grain as a prize. Interestingly, the birds adjusted their probability of switching keys or staying to approximate the optimal strategy. Replication of the procedure with human participants showed that humans failed to adopt optimal strategies, even with extensive training.

[1] See also the Two Envelope Problem (Chapter ///).

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