Repeated Trials and Sampling

Abstract

This chapter studies a mathematical model for repeated trials, each of which may result in some event either happening or not happening. Occurrence of the event is called success, and non-occurrence called failure. For instance: Suppose that on each trial there is success with probability p, failure with probability q = 1−p, and assume the trials are independent. Such trials are called Bernoulli trials or Bernoulli (p) trials to indicate the success probability p. The number of successes in n trials then cannot be predicted exactly. But if n is large we expect the number of successes to be about np, so the relative frequency of successes will, most likely, be close to p. The important questions treated in this chapter are: how likely? and how close? The answers to these questions, first discovered by the mathematicians James Bernoulli and Abraham De Moivre, around 1700, are the mathematical basis of the long-run frequency interpretation of probabilities.