The History of Mathematics: A Brief Course

518 18. PROBABILITY AND STATISTICS

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Figure 1. Frequencies of numbers in a state lottery over a one-

year period.

could imagine the bell-shaped normal curve that we are familiar with, but he could

not give it an equation. Instead he described it as the curve whose ordinates were

numbers having certain logarithms. What seems most advanced in his analysis is

that he recognized the area under the curve as a probability and computed it by

a mechanical quadrature method that he credited jointly to Newton, Roger Cotes,

James Stirling, and himself. This tendency of the average of many independent

trials to look like the bell-shaped curve is called the central limit theorem.

It is difficult to appreciate the work of Bernoulli and de Moivre in applications

without seeing it applied in a real-world illustration. To take a very simple example,

consider Fig. 1, which is a histogram of the frequencies with which the numbers

from 1 to 42 were drawn in a state lottery over a period of one year^8 Six numbers

are drawn twice a week, for a total of 624 numbers each year. At each drawing a

given number has a probability of | of being drawn. Thus, focusing attention only

on the occurrence of a fixed integer fc, we can think of the lottery as a series of 104

independent trials with a probability of success (drawing the number fc) equal to \

at each trial.

Although the individual data do not reveal the binomial distribution or show

any bell-shaped curve, we can think of the frequencies with which the 42 numbers

are drawn as the data for a second probabilistic model. By the binomial distribu-

tion, for each frequency r from 0 to 104, The probability that a given number will

be drawn r times should theoretically be

If the probability of an event is proportional to the number of times that the
event occurs in a large number of trials, then the number of numbers drawn r
times should be 42 times this expression. The resulting theoretical frequencies are
negligibly small for r < 6 or r > 23. The values predicted by this theoretical model
for r between 6 and 23, rounded to the nearest integer, are given in the second row
of the following table, while the experimentally observed numbers are given in the
bottom row.

(^8) The Tri-state Megabucks of Maine, New Hampshire, and Vermont, from mid-December 2000 to
mid-December 2001.