Chapter 7 Probability

Definition of Probability What is probability? There seems to be no agreement on the answer. There are two broad schools of thought: frequency and nonfrequency. Even among the frequency school, there are at least two definitions: a priori and a posteriori. A priori is defined without any empirical data. It is rather a statement about one’s state of mind. It is the basis of theoretical mathematical probability.

Definition of Probability A posteriori, or relative long-run frequency, definition is empirical nature. It says that, in an actual series of tests, probability is the ratio of the number of times an event occurs to the total number of trials. A priori approach supplies a constitutive definition of probability, whereas the a posteriori approach supplies an operational definition of probability.

Definition of Probability In nonfrequency approach, there are two values: (1) the probability value itself, and (2) the weight of evidence associated with it. The weight of evidence is subjective. In summary, the long-run relative frequency approach is the most prevalent in behavioral science research.

Sample Space, Sample Points, and Events The sample space includes all possible outcomes of an “experiment” that are of interest to the experimenter. The primary elements of U are called elements or sample points. Table 7.1, Figure 7.1, 7.2 An event is a subset of U. In standard usage, events are more encompassing than points. All points are events (subsets), but not all events are points.

Determining Probabilities with Coins The probabilities of all the points in the sample space must add up to Figure 7.3. Each complete path of the tree (from the start to the third toss) is a sample point.

An Experiment with Dice Table 7.2

Some Formal Theory A weight is a positive number assigned to each each element, x, in U, and written w(x), such that the sum of all these weights,, is equal to 1. We write m(A), meaning “The measure of the set A.” This simply says the sum of the weights of the elements in the set A. For example,

Compound Events and Their Probabilities An event is a set of possibilities; it is a possible set of events; it is an outcome of a probability “experiment.” A compound event is the co-occurrence of two or more single (or compound) events. The two set operations of intersection and union—the operations of most interest to us—imply compound events.

Compound Events and Their Probabilities If there is no overlap between two sets,, then the following equation holds: If two sets are not disjoint, rather they overlap,, then the following equation holds:

Independence, Mutual Exclusiveness, and Exhaustiveness Exhaustiveness means that the subsets of U use up all the sample space, or, where A, B, …,K are subsets of U, the sample space. In probability language, this means: If the events (sets) A, B, and C are mutually exclusive, then

Independence, Mutual Exclusiveness, and Exhaustiveness Two events, A and B, are statistically independent if the following equation holds: Suppose we rank order examination papers and then assign grades on the basis of these ranks, the grades given by the rank-order method are not independent. In any area of research, one cannot assume that multiple observations of one subject are independent.

Conditional Probability Definition of Conditional Probability Let A and B be events in the same space, U, as usual. The conditional probability is denoted: p(A ︳ B), which is read, “The probability of A, given B.” The formula for the conditional probability involving two events is: The sample space has, through knowledge, been reduced from U to B.

Conditional Probability When events are independent, Table 7.5 The probability of success without any other knowledge is a probability problem on the whole sample space U. This probability is 0.4. But given knowledge of MAT score, the sample space is reduced from U to a subset of Figure 7.5

Bayes’ Theorem: Revising Probabilities With Bayes’ Theorem, one could update or revise current probabilities based on new information or data.

Bayes’ Theorem: Revising Probabilities