172 CHAPTER 4 Probability Manufacturing EXAMPLE 1 A factory of the Global Manufacturing Company was manufacturing products with a defect rate of 15% (based on data from the Harvard Business Review). If a customer purchases 12 of the products, what is the probability of getting at least one that is defective? YOUR TURN. Do Exercise 7 “Births in the United States.” SOLUTION Step 1: Let A = at least 1 of the 12 products is defective. Step 2: Identify the event that is the complement of A. A = not getting at least 1 defect among the 12 items = all 12 items are good with no defects Step 3: Find the probability of the complement by evaluating P1A2. P1A2 = P1all 12 items are good2 = 0.85# 0.85# 0.85# 0.85# 0.85# 0.85# 0.85# 0.85# 0.85# 0.85# 0.85# 0.85 = 0.8512 = 0.142 Step 4: Find P1A2 by evaluating 1 - P1A2. P1A2 = 1 - P1A2 = 1 - 0.142 = 0.858 INTERPRETATION For a group of 12 products, there is a 0.858 probability of getting at least 1 that is defective. As quality control goes, this is too high. The company needs to improve its manufacturing process so that the defect rate is lowered. We now consider the principle that the probability of an event is often affected by knowledge that some other event has occurred. According to Golf Digest, the probability of a golfer making a hole in one is 1>12,500, but if you have the additional information that the golfer is a professional, the probability changes to 1>2500. In general, a conditional probability of an event is used when the probability is calculated with some additional knowledge, such as the knowledge that some other event has occurred. (Conditional probabilities were used in Section 4-2 with situations in which samples were selected without replacement.) PART 2 Conditional Probability DEFINITION A conditional probability of an event is a probability obtained with the additional information that some other event occurred. Notation P1B A2 denotes the conditional probability of event B occurring, given that event A has already occurred. INTUITIVE APPROACH FOR FINDING P1 B∣ A2 The conditional probability of B occurring given that A has occurred can be found by assuming that event A occurred and then calculating the probability that event B will occur, as illustrated in Example 2. Convicted by Probability A witness described a Los Angeles robber as a Caucasian woman with blond hair in a ponytail who escaped in a yellow car driven by an AfricanAmerican male with a mustache and beard. Janet and Malcolm Collins fit this description, and they were convicted based on testimony that there is only about 1 chance in 12 million that any couple would have these characteristics. It was estimated that the probability of a yellow car is 1>10, and the other probabilities were estimated to be 1>10, 1>3, 1>10, and 1>1000. The convictions were later overturned when it was noted that no evidence was presented to support the estimated probabilities or the independence of the events. Also, because the couple was not randomly selected, a serious error was made in not considering the probability of other couples being in the same region with the same characteristics. A d L r C w b
RkJQdWJsaXNoZXIy NjM5ODQ=