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As a Las Vegas aficionado, I'm always glad to see serious attempts to dispel
popular misunderstandings in the realm of probability and statistics. I must
admit that I'm at best indifferent about anyone actually reading them,
though. It makes my job as a poker player harder, and I win far fewer drinks
from my friends in bars.
But, just as Las Vegas was built by people who are good at math for people who are bad at math, there is the very real likelihood that the general public can be led astray by intentional or unintentional misrepresentations of probabilities. Author John Haigh's goal in writing Taking Chances is to reduce the likelihood of those misunderstandings and to educate readers as to the true probabilities involved in buying lottery tickets or in the real probability that DNA evidence points to an individual defendant.
Some of the distinctions Haigh discusses are quite subtle, and even the sophisticated can be caught unawares. Other problems are relatively straightforward but run counter to common sense. One example of such a problem is what is known in the U.S. as the Monte Hall problem. Monte Hall was the host of Let's Make a Deal, where one of the games had Monte showing a player three doors. Behind one of the three doors was a great prize, but the other two doors hid worthless prizes, such as a carrot. In this game, Monte knows which door concealed the big prize. After the contestant picked a door, Monte would show one of the losing doors and ask the contestant if they wanted to switch the door they chose first for the remaining door.
The question is, is it better to switch, to keep the original door, or are both strategies equal? Common sense says that there are two doors remaining, so the probability you have picked the winner is 1/2. If that's the case, then it makes no difference whether you switch or not. The real answer, though, relies on the fact that Monte knows which door conceals the grand prize. He will always reveal a losing door (you can't pick both of them), so you have to look back to the only reliable information you have, which is the probability that you picked the winning door. That probability is 1/3, so by switching you have a 2/3 chance of winning. If you don't believe it, have a friend arrange three cards and see how never switching plays out.
The author assumes the reader has a solid grounding in math, but there are tutorials on the calculations and counting methods in the appendices. In the end, with an inexpensive book such as Taking Chances available for under $20, if you remain ignorant of probability you have only yourself to blame.