4-2 Addition Rule and Multiplication Rule 165 WATCH YOUR LANGUAGE! Example 6 illustrates that finding correct or relevant probability values often requires greater language skills than computational skills. In Example 6, what exactly do we mean by “same day of the week”? See how parts (a) and (b) in Example 6 are very different. Redundancy: Important Application of Multiplication Rule The principle of redundancy is used to increase the reliability of many systems. Our eyes have passive redundancy in the sense that if one of them fails, we continue to see. An important finding of modern biology is that genes in an organism can often work in place of each other. Engineers often design redundant components so that the whole system will not fail because of the failure of a single component, as in the following example. Using Probability for Significance EXAMPLE 7 Let’s now revisit the Chapter Problem and apply what we have learned about independence and the multiplication rule. The Chapter Problem describes how the XSORT method of gender selection was developed to help couples have baby girls when they have children. If a test of the XSORT method of gender selection involves 20 couples who give birth to 20 babies, what is the probability that the 20 babies are all girls? Does the result of 20 girls suggest that the XSORT technique is effective? YOUR TURN. Do Exercise 4 “Probability for Significance.” SOLUTION We assume that the XSORT method has no effect so that P1girl2 = 1>2. Because the 20 births are all independent, we can apply the multiplication rule as follows: P120 girls2 = P11st baby is a girl2 # P12nd baby is a girl2 # c# P120th baby is a girl2 = 1 2 # 1 2 # 1 2 # c# 1 2 120 times2 = a 1 2b 20 = 0.000000954 Notice that this result of 0.000000954 matches the probability given for the Test A Result (probability of 20 girls born by chance) in Figure 4-1 on page 145. We love it when a plan comes together! Now we must make a decision: Is the result of 20 girls in 20 births likely to occur with random chance, or does it appear that the XSORT method is effective in enabling couples to have baby girls? Because the probability of getting 20 girls by chance (with no effect from the XSORT method) is only 0.000000954, it makes sense to eliminate chance as a viable explanation. The result of 20 girls in 20 births is a significantly high number of girls (probability 0.05 or less), so it does appear that the XSORT method is effective. CP To Win, Bet Boldly The New York Times published an article by Andrew Pollack in which he reported lower than expected earnings for the Mirage casino in Las Vegas. He wrote that “winnings for Mirage can be particularly volatile, because it caters to high rollers, gamblers who might bet $100,000 or more on a hand of cards. The law of averages does not work as consistently for a few large bets as it does for thousands of smaller ones. . .” This reflects the most fundamental principle of gambling: To win, place one big bet instead of many small bets! With the right game, such as craps, you have just under a 50% chance of doubling your money if you place one big bet. With many small bets, your chance of doubling your money drops substantially.
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