Tutorial on probability theory

Abstract

Probability measures the amount of uncertainty of an event: a fact whose occurrence is uncertain. Consider, as an example, the event R “Tomorrow, January 16th, it will rain in Amherst”. The occurrence of R is difficult to predict — we have all been victims of wrong forecasts made by the “weather channel” — and we quantify this uncertainty with a number p(R), called the probability of R. It is common to assume that this number is non-negative and it cannot exceed 1. The two extremes are interpreted as the probability of the impossible event: p(R) = 0, and the probability of the sure event: p(R) = 1. Thus, p(R) = 0 asserts that the event R will not occur while, on the other hand, p(R) = 1 asserts that R will occur with certainty. Suppose now that you are asked to quote the probability of R, and your answer is p(R) = 0.7. There are two main interpretations of this number. The ratio 0.7/03 represent the odds in favor of R. This is the subjective probability that measures your personal belief in R. Objective probability is the interpretation of p(R) = 0.7 as a relative frequency. Suppose, for instance, that in the last ten years, it rained 7 times on the day 16th January. Then 0.7 = 7/10 is the relative frequency of occurrences of R, also given by the ratio between the favorable cases (7) and all possible cases (10). There are other interpretations of p(R) = 0.7 arising, for instance, from logic or psychology (see Good (1968) for an overview.) Here, we will simply focus attention to rules for computations with probability.