Inevitable Probabilities: Two Fascinating Mathematical Results
ALEA IACTA EST, OVER AND OVER
In the beginning of Chapter 5, I mentioned how consecutive averages of repeated die rolls settle in toward the expected value 3.5 as the number of rolls increases. When I wrote this, I expected you to accept it as reasonable or perhaps, if you are the suspicious type, try it out for yourself. And early in Chapter 1, I gave an interpretation of the probability of heads being 0.5: that you can expect to get heads about 50% of the time in a large number of coin tosses. The proportion of heads settles in toward the probability of heads. It seems that both probabilities and expected values can be interpreted in this way in terms of average long-term behavior, at least in these examples. It is easy to believe that such an interpretation is always possible, and it is certainly nice to have our theory solidly anchored in the real world in this way. Now that we have become sophisticated, this kind of vague and intuitive reasoning is not enough though. We want proof, unquestionable mathematical proof, that averages approach expected values and that proportions approach probabilities. Luckily, this can be done.
That is, it can be done if we can make some reasonable assumptions, for example, that the coin is not tossed by probabilist and magician Persi Diaconis who has the unusual ability to make a coin land heads every time (and can probably make it disappear too).1 But once we have adequately described ...