Monday, August 08, 2016

The Law of Large Numbers

John E. Freund (Introduction to Probability) has been discussing topics like the "odds" that an airline flight from Chicago to Los Angeles will arrive on time. He says that if 688 of the last 800 flights have been on time, we can say the probability of this flight being on time is .86.

Then he asks, "When probabilities are thus estimated, it is only reasonable to ask whether the estimates are any good. The answer, which is 'Yes,' is supported by a remarkable law called the Law of Large Numbers... Informally, this law can be stated as follows:

"If the number of times the situation is repeated becomes larger and larger, the proportion of successes this will tend to come closer and closer to the actual probability of success."

Later on, he states this law formally:

"If a random variable has the binomial distribution, the probability is at least 1 - 1 / k2 that the proportion of successes in n trials will differ from p by less than k * (p(1 - p) / n)1/2."

What I wish to draw your attention to here is that the formal definition of the law refers entirely to elements of a mathematical model, and not to the real world. There exists a particular "actual probability of success" only because we have stipulated that we are dealing with a "random variable" that has a binomial distribution with a known value for p.

But events in the real world are not determined by "random variables," but by snowstorms and bomb threats and pilots calling in sick. (And if you protest, "But, those are random variables!" you're making the same mistake as Freund.) A random variable is an element of a mathematical-scientific model, not of the real world. And whether real world events in some particular situation are closely mirrored by some model employing a random variable was precisely the question Freund was supposed to be answering in the first place. Certainly, the question of whether some model closely mirrors the real world can't be answered by exploring the formal features of the model itself!

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