Notes

Probability - the workhorse for inference (Starbird)

The long run:

A great many of the things that happen in this world appear to occur very erratically if one looks only at a few instances. Over a very wide range, however, these erratic phenomena "smooth out" and show a more and more regular behavior as the number of instances of occurrence, or of trial, is increased.

You can furnish all sorts of examples of this from your own immediate experience. If you make a record of the occurrence of heads and tails in tossing a coin, you are bound to have all one or all the other after a single toss. After five tosses there may well be three of one kind and two of the other, and you will not be particularly surprised if there are four of one kind and only one of the other. But after 100 tosses you will expect to have not too far from half of each.

You would be pretty surprised to have a four-to-one mixture of heads to tails (i.e., 80 heads and 20 tails) after 100 tosses. And if you had 80 per cent heads and 20 per cent tails after 1000 tosses you would justifiably examine the coin very carefully; on the average, only once in about a million billion billion billion trials, each of 1000 tosses, would you get a discrepancy - from an even break between heads and tails - as large or larger! than 800 to 200.

It would be interesting but trivial if this sort of behavior were exhibited only by coins and dice. The fact is, however, that births, deaths, and a very large number of the important human experiences between birth and death, when measured, or assigned numerical value or rank, or tabulated, exhibit behavior most usefully illuminated by the theorems which probability furnishes.

Since I appeal to your intuition and common sense in considering the way coin-tossing proceeds, perhaps you are thinking that intuition and common sense are enough, that you really could "guess" a sensible answer to any problem of this sort. This happens to

be spectacularly not the case, as you will see in this and the following two chapters. Probability theory not only refines and makes quantitatively precise a good many ideas which do seem intuitively reasonable; it also reveals many things that intuitively

seem very astonishing indeed.

In fact, people often use the phrase "the law of averages" to justify incorrect remarks.

References: Warren Weaver, "Lady Luck", Chapter 7.