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Discrete Probability

As an illustration of the applications of some of the concepts and tools studied, we present a brief introduction to discrete probability theory. We recall that an experiment is a physical process that has a number of observable outcomes. We note that dealing a poker hand has C(52, 5) = 2,598,960 possible outcomes; and examining a student’s transcript has 54 possible outcomes (assuming that the student takes four courses, and the five possible grades are A, B, C, D, F). In our model of a physical process, these outcomes are considered to be mutually exclusive and exhaustive; that is, exactly one outcome will take place in any particular instance of the experiment. Thus, when we toss a coin, either a head or a tail will show. It is not possible that both of them will show, nor is it possible that none of them will show. (If indeed, we believe that the coin might stand on its side, e should include in our model three possible outcomes when a coin is tossed – namely, head, tail, and standing on its side.)

Formally, we refer to the set of all possible outcomes of an experiment as the sample space of the experiment. We also refer to the outcomes in the sample space as samples or sample points. We shall use the notation S = {x₁, x₂, …, x_i, ….} for a sample space S consisting of the samples x₁, x₂, …., x_i, and so on. A sample space that has a finite number or a countably infinite number of samples is called a discrete sample space. We shall restrict our discussion to discrete sample spaces. For example, for the experiment of tossing a coin, the sample space is a set S = {h, t}, which consists of the two possible outcomes h (head) and t (tail). For the experiment of tossing two coins, the sample space is a set S = {hh, ht, th, tt}, which consists of the four possible outcomes head-head, head-tail, tail-head and tail-tail. Also, for the experiment of waiting for the arrival of a bus at a bus stop, the sample space is a set S = {0, 1, 2, 3, …, 30} in which the outcomes are the waiting times ranging from 0 to 30 minutes. Consider the experiment of shooting at a target until there is a hit. The sample space is a countably infinite set S = {h, mh, mmh, mmmh, ….}, where h denotes a hit, mh denotes a miss followed by a hit, mmh denotes two misses followed by a hit, and so on.

Associated with each sample in sample space is a real number called the probability of that sample? For the sample x, we shall use p(x_i) to denote the probability associated with x_i. The probabilities associated with the samples must satisfy two conditions:

1. The probability of each sample is a nonnegative number less than or equal to 1. That is, for each x_i in S, 0 ≦ p(x_i) ≦ 1.

2. The sum of the probabilities of all the samples in the sample space is equal to 1. That is, Σ_xiϵS p(x_i) = 1.