Drunkard's Walk - Leonard Mlodinow [28]
One has to give credit to Marilyn vos Savant, not only for attempting to raise public understanding of elementary probability but also for having the courage to continue to publish such questions even after her frustrating Monty Hall experience. We will end this discussion with another question taken from her column, this one from March 1996:
My dad heard this story on the radio. At Duke University, two students had received A’s in chemistry all semester. But on the night before the final exam, they were partying in another state and didn’t get back to Duke until it was over. Their excuse to the professor was that they had a flat tire, and they asked if they could take a make-up test. The professor agreed, wrote out a test, and sent the two to separate rooms to take it. The first question (on one side of the paper) was worth five points. Then they flipped the paper over and found the second question, worth 95 points: “which tire was it?” What was the probability that both students would say the same thing? My dad and I think it’s 1 in 16. Is that right?14
No, it is not: If the students were lying, the correct probability of their choosing the same answer is 1 in 4 (if you need help to see why, you can look at the notes at the back of this book).15 And now that we’re accustomed to decomposing a problem into lists of possibilities, we are ready to employ the law of the sample space to tackle the Monty Hall problem.
AS I SAID EARLIER, understanding the Monty Hall problem requires no mathematical training. But it does require some careful logical thought, so if you are reading this while watching Simpsons reruns, you might want to postpone one activity or the other. The good news is it goes on for only a few pages.
In the Monty Hall problem you are facing three doors: behind one door is something valuable, say a shiny red Maserati; behind the other two, an item of far less interest, say the complete works of Shakespeare in Serbian. You have chosen door 1. The sample space in this case is this list of three possible outcomes:
Maserati is behind door 1.
Maserati is behind door 2.
Maserati is behind door 3.
Each of these has a probability of 1 in 3. Since the assumption is that most people would prefer the Maserati, the first case is the winning case, and your chances of having guessed right are 1 in 3.
Now according to the problem, the next thing that happens is that the host, who knows what’s behind all the doors, opens one you did not choose, revealing one of the sets of Shakespeare. In opening this door, the host has used what he knows to avoid revealing the Maserati, so this is not a completely random process. There are two cases to consider.
One is the case in which your initial choice was correct. Let’s call that the Lucky Guess scenario. The host will now randomly open door 2 or door 3, and, if you choose to switch, instead of enjoying a fast, sexy ride, you’ll be the owner of Troilus and Cressida in the Torlakian dialect. In the Lucky Guess scenario you are better off not switching—but the probability of landing in the Lucky Guess scenario is only 1 in 3.
The other case we must consider is that in which your initial choice was wrong. We’ll call that the Wrong Guess scenario. The chances you guessed wrong are 2 out of 3, so the Wrong Guess scenario is twice as likely to occur as the Lucky Guess scenario. How does the Wrong Guess scenario differ from the Lucky Guess scenario? In the Wrong Guess scenario the Maserati is behind one of the doors you did not choose, and a copy of the Serbian Shakespeare is behind the other unchosen door. Unlike the Lucky Guess scenario, in this scenario the host does not randomly open an unchosen door. Since he does not want to reveal the Maserati, he chooses to open precisely the door that does not have the Maserati behind it. In other words, in the Wrong Guess scenario the host intervenes in what until now has been a random process. So the process is no longer random: the host uses his knowledge to bias