Probability and Applications

Transcription

1 Chapter 5 Probability and Applications 5.2 SOME USEFUL DEFINITIONS Random experiment: a process that has an unknown outcome or outcomes that are known only after the process is completed. Event: an outcome of a random experiment. Sample space: a collection of all possible outcomes of a random experiment (denoted S). Simple event: a single point within the sample space. Compound event: a collection of simple events. 5.3 PROBABILITY SOURCES Objective Probability Classical (objective) probability: if a random experiment has N equally likely outcomes and N A of them are favorable to event A, then the probability of event A is PA N A =N: Relative frequency as a probability: if a random experiment is repeated a large number (n) of times and event A is observed in N A of these n repetitions, then the relative frequency of event A is n A /n. Moreover, if n A /n approaches some long-run stable value P(A) as n!1, then lim n!1 n A =n PA is called the probability of event A. 5.4 SOME USEFUL DEFINITIONS INVOLVING SETS OF EVENTS IN THE SAMPLE SPACE Suppose A and B are events within the sample space S. Introduction to Quantitative Methods in Business: With Applications Using Microsoft Office Excel, First Edition. Bharat Kolluri, Michael J. Panik, and Rao Singamsetti John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. 96

2 5.5 Probability Laws 97 Union of events A and B: the set of simple events that are either in A or in B or in both A and B. Intersection of events A and B: the set of simple events that are common to events A and B. Complement of event A (denoted A): consists of all simple events within S that are not in A. Mutually exclusive events: events A and B are mutually exclusive if A B ϕ. Thus, the occurrence of A precludes the occurrence of B and vice versa. 5.5 PROBABILITY LAWS For events A and B within S: General additional rule: PA B PA PB PA B.Here,P(A) and P(B) are marginal probabilities and PA B is a joint probability. Special additional rule: ifa Bϕ, then PA B PA PB Rule of Complements According to the Rule of Complements, P A 1 PA, that is, the probability that event A does not occur equals 1 minus the probability that event A does occur. This is because A A S and PS 1: Conditional Probability Suppose an event B has definitely occurred. What is the probability that an event A has also occurred? The probability of A given B is called the conditional probability of A given B and written PAjB, where is read given. In general, PAjB PBjA PA B PB PA B PA ; PB 0; ; PA 0: General Multiplication Rule (Product Rule) From the definition of the conditional probabilities PAjBand PBjA,wecan write the general multiplication rule as PA B PAjB PB PBjA PA

3 98 Chapter 5 Probability and Applications since PA B appears in the numerator of both of these conditional probabilities Independent Events Two events A and B are independent if the occurrence of one of them in no way affects the probability of occurrence of the other. Hence, A and B are independent if and only if PAjB PA and PBjA PB. For independent events, we have the special multiplication rule: PA B PA PB : What is the distinction between events that are mutually exclusive and events that are independent? Mutually exclusive events cannot occur together, that is, A B cannot occur. Thus, A B ϕ and thus PA B 0. Independent events can occur together: It is just that when one occurs, it does not affect the probability of occurrence of the other. Hence, A B ϕ and thus PA B 0. In fact, PA B PA PB Probability Tree Approach This approach is useful when we have multiple trials of a random experiment. For instance, suppose we have two possible outcomes for our random experiment (events A and B) and that PAPB 1. Suppose also that three separate trials are conducted. From an initial node, the tree starts with two branches representing the occurrence of either A or B on the first trial (Figure 5.1). From each of these two nodes, we branch again for trial two. And on trial three, we branch from each of the four nodes determined on trial two. We finish with eight terminal nodes. For instance, under three independent trials, we branch as indicated in Figure CONTINGENCY TABLE A contingency table is an array of data arranged in rows and columns. It is a useful device for determining probabilities. For example, given two row categories (R 1 and R 2 ) and column categories (C 1 and C 2 ), Table 5.1 houses the distribution of n = 130 data values. Hence, we may compute PR 1 C ; PC ; PC 1jR 2 PC 1 R 2 PR 2 10=130 10=60; and so on 60=130 Table5.1canbetransformedintoaprobability table (Table 5.2) by dividing each of its entries by 130.

4 Table 5.1 Contingency Table 5.6 Contingency Table 99 C 1 C 2 Total R R Total Figure 5.1 Probability tree.

5 100 Chapter 5 Probability and Applications Table 5.2 Probability Table C 1 C 2 Total R 1 30/130 40/130 70/130 R 2 10/130 50/130 60/130 Total 40/130 90/130 1