Drunkard's Walk - Leonard Mlodinow [29]
To summarize: if you are in the Lucky Guess scenario (probability 1 in 3), you’ll win if you stick with your choice. If you are in the Wrong Guess scenario (probability 2 in 3), owing to the actions of the host, you will win if you switch your choice. And so your decision comes down to a guess: in which scenario do you find yourself? If you feel that ESP or fate has guided your initial choice, maybe you shouldn’t switch. But unless you can bend silver spoons into pretzels with your brain waves, the odds are 2 to 1 that you are in the Wrong Guess scenario, and so it is better to switch. Statistics from the television program bear this out: those who found themselves in the situation described in the problem and switched their choice won about twice as often as those who did not.
The Monty Hall problem is hard to grasp because unless you think about it carefully, the role of the host, like that of your mother, goes unappreciated. But the host is fixing the game. The host’s role can be made obvious if we suppose that instead of 3 doors, there were 100. You still choose door 1, but now you have a probability of 1 in 100 of being right. Meanwhile the chance of the Maserati’s being behind one of the other doors is 99 in 100. As before, the host opens all but one of the doors that you did not pick, being sure not to open the door hiding the Maserati if it is one of them. After he is done, the chances are still 1 in 100 that the Maserati was behind the door you chose and still 99 in 100 that it was behind one of the other doors. But now, thanks to the intervention of the host, there is only one door left representing all 99 of those other doors, and so the probability that the Maserati is behind that remaining door is 99 out of 100!
Had the Monty Hall problem been around in Cardano’s day, would he have been a Marilyn vos Savant or a Paul Erdös? The law of the sample space handles the problem nicely, but we have no way of knowing for sure, for the earliest known statement of the problem (under a different name) didn’t occur until 1959, in an article by Martin Gardner in Scientific American.16 Gardner called it “a wonderfully confusing little problem” and noted that “in no other branch of mathematics is it so easy for experts to blunder as in probability theory.” Of course, to a mathematician a blunder is an issue of embarrassment, but to a gambler it is an issue of livelihood. And so it is fitting that when it came to the first systematic theory of probability, it took Cardano, the gambler, to figure things out.
ONE DAY while Cardano was in his teens, one of his friends died suddenly. After a few months, Cardano noticed, his friend’s name was no longer mentioned by anyone. This saddened him and left a deep impression. How does one overcome the fact that life is transitory? He decided that the only way was to leave something behind—heirs or lasting works of some kind or both. In his autobiography, Cardano describes developing “an unshakable ambition” to leave his mark on the world.17
After obtaining his medical degree, Cardano returned to Milan, seeking employment. While in college he had written a paper, “On the Differing Opinions of Physicians,” that essentially called the medical establishment a bunch of quacks. The Milan College of Physicians now returned the favor, refusing to admit him. That meant he could not practice in Milan. And so, using money he had saved from his tutoring and gambling, Cardano bought a tiny house to the east, in the town of Piove di Sacco. He expected to do good business there because disease was rife in the town and it had no physician. But his market research had a fatal flaw: the town had no doctor because the populace preferred to be treated by sorcerers and priests. After years of intense work and study, Cardano found himself with little income but a