Appendix B clarifies what the law of large numbers is, why it does not apply to
the psychological research on sample size, and which mathematical results do
apply.

Simeon Denis Poisson ( 1837) was the first to introduce the term 'law of large
numbers' for Bernoulli's theorem, which had been published posthumously in
Ars Conjectandi ( 1713). In modern notation, Bernoulli's version of the theorem
can be stated as follows ( Stigler, 1986, p. 66). Suppose an experiment with two
possible outcomes is to be repeated many times. If p is the probability of suc-
cess in any single experiment, and if non-negative numbers ε and c are specified,
then the number of trials n can be determined such that the number of observed
successes m in n trials satisfies
. (1)
The setup described above is known today as a ' Bernoulli process'. Bernoulli
himself used an urn model with r 'fertile' and s 'sterile' equally likely cases so
that p = r / (r + s). He set ε equal to 1 / (r + s) and proposed making c large
enough to ensure 'moral certainty'. Bernoulli calculated the number of trials re-
quired for the case in which r = 30 and s = 20 and, because he had high standards
of moral certainty, for c = 1,000, 10,000, and 100,000 ( Bernoulli, 1713, p.
238). For c = 1000--where the probability P of m / n falling within the inter-
val [29/50, 31/50] is at least 1000 times larger than the probability of m / n
falling outside of that interval--he calculated that he would need at least n =
25,550 observations. This discouragingly large number might have been one
reason for the abrupt conclusion of his Ars Conjectandi ( Stigler, 1986, p. 77).

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