180 CHAPTER 4 Probability 1. Multiplication Counting Rule The multiplication counting rule is used to find the total number of possibilities from some sequence of events. 22.Redundancy in Stadium Generators Large stadiums rely on backup generators to provide electricity in the event of a power failure. Assume that emergency backup generators fail 22% of the times when they are needed (based on data from Arshad Mansoor, senior vice president with the Electric Power Research Institute). A stadium has three backup generators so that power is available if at least one of them works in a power failure. Find the probability of having at least one of the backup generators working given that a power failure has occurred. Does the result appear to be adequate for the stadium’s needs? 23.Composite Drug Test Based on the data in Table 4-1 on page 159, assume that the probability of a randomly selected person testing positive for drug use is 0.126. If drug screening samples are collected from 5 random subjects and combined, find the probability that the combined sample will reveal a positive result. Is that probability low enough so that further testing of the individual samples is rarely necessary? 24. Composite Water Samples The Fairfield County Department of Public Health tests water for the presence of E. coli (Escherichia coli) bacteria. To reduce laboratory costs, water samples from 10 public swimming areas are combined for one test, and further testing is done only if the combined sample tests positive. Based on past results, there is a 0.005 probability of finding E. coli bacteria in a public swimming area. Find the probability that a combined sample from 10 public swimming areas will reveal the presence of E. coli bacteria. Is that probability low enough so that further testing of the individual samples is rarely necessary? 25.Shared Birthdays Find the probability that of 25 randomly selected people, at least 2 share the same birthday. 26.Unseen Coins A statistics professor tosses two coins that cannot be seen by any students. One student asks this question: “Did one of the coins turn up heads?” Given that the professor’s response is “yes,” find the probability that both coins turned up heads. 4-3 Beyond the Basics Key Concept Probability problems typically require that we know the total number of simple events, but finding that number often requires one of the five rules presented in this section. In Section 4-2 with the addition rule, multiplication rule, and conditional probability, we encouraged intuitive rules based on understanding and we discouraged blind use of formulas, but this section requires much greater use of formulas as we consider five different methods for counting the number of possible outcomes in a variety of situations. Not all counting problems can be solved with these five methods, but they do provide a strong foundation for the most common real applications. 4-4 Counting MULTIPLICATION COUNTING RULE For a sequence of events in which the first event can occur n1 ways, the second event can occur n2 ways, the third event can occur n3 ways, and so on, the total number of outcomes is n1 # n2 # n3 c.
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