Elements of Probability and Statistics
In this appendix, we offer a brief survey of the parts of probability theory and its application to statistics that are most relevant to psychophysics. Because our aim is to make references to these ideas in the body of book more comprehensible, we frequently allude to psychophysical applications. Most concepts would be covered in a one-semester behavioral statistics course, but some ideas (e.g., random variable) are not usually found at that level. We do not believe, of course, that we have exhausted in this brief chapter topics usually covered in a semester or two. Our incomplete discussion is also relatively informal. Hays (1994) provided a thorough treatment of all issues raised here.
Definition of Probability. In the simplest probabilistic situation, an elementary event is chosen at random from the sample space S. If A is a subset of S, then the probability that an elementary event that is in A will occur iswhere the function n counts the number of elementary events in a set. For example, when a fair coin is tossed, the probability of a Head occurring is 1/2, as is the probability of a Tail. When a die is tossed, the probability of a “2” is 1/6, and the probability of an Even outcome (“2, ” “4, ” or “6”) is 3/6 or 1/2.
Some important characteristics of probabilities are evident from these examples: All probabilities must lie between 0 and 1, and the sum of probabilities for all elementary events must be 1.