bj

Previously, we showed that if two events are mutually exclusive, then the probability of their union is the sum of the probabilities of each event. POAUB) - PA+PB Next, we want to determine the result when events A and B are not mutually exclu- sive. In Section 3.1 we noted that events A and n Bare mutually exclusive and thus, PAUB = PLA) Pna) In addition, events in Band n Bare mutually exclusive, and their union is B. P(B) = PAN BU PB) From this we can derive the following result PAB) - PB - POBI Combining these two results, we obtain the addition rule of probabilities, as shown in Figure 3.7. The Addition Rule of Probabilities Let A and B be two events. Using the addition rule of probabilities, the probability of their union is as follows: PLAUB) = P(A) + PCB) - PAB (3.8) The Venn diagram in Figure 37 provides an intuitive understanding of the addition rule The larger rectangle, S. represents the entire simple space. The smaller circles, A and B repre sent events and B. We can see that the area where A and B overlap presents the intersec tion of the two probabilities, PAB). To compute the probability of the union of events A and B, we first add the events probabilities, PAI + P(B). However, notice that the prob- ability of the intersection, Plan B), is counted twice and thus must be subtracted once Figure 3.7 Venn Diagram for Addition Rule P(AUB) = PLA) + PLB) - PAB) PLAUB) 913 PA PBY PA + Q Example 3.14 Product Selection (Addition Rule) A cell phone company found that 75% of all customers want text messaging on their phones, 80% want photo capability, and 65% want both. What is the probability that a customer will want at least one of these? 92 Chapter Probability