SECTION 3.1 Basic Concepts of Probability and Counting 135 When an experiment is repeated many times, regular patterns are formed. These patterns make it possible to find empirical probability. Empirical probability can be used even when each outcome of an event is not equally likely to occur. Empirical (or statistical) probability is based on observations obtained from probability experiments. The empirical probability of an event E is the relative frequency of event E. P1E2 = Frequency of event E Total frequency = f n Note that n = gf. DEFINITION Finding Empirical Probabilities A research organization is conducting a survey of randomly selected U.S. adults to determine how they read books during the past year, if at all. So far, 1502 adults have been surveyed. The pie chart shows the results. (Note that digital books include ebooks as well as audio books.) What is the probability that the next adult surveyed read only print books during the last year? (Adapted from Pew Research Center) Read only print books Read no books Read both print and digital books Read only digital books 91 Book Reading by U.S. Adults 108 560 425 409 SOLUTION Note that the responses are not equally likely to occur and are based on observations. So, you cannot use the formula for classical probability, but you can use the formula for empirical probability. The event is a response of “read only print books.” The frequency of this event is 560. The total of the frequencies is n = 560 + 108 + 425 + 409 Add frequency of each response. = 1502. Total frequency The empirical probability that the response of the next adult is “read only print books” is P(read only print books) = 560 1502 Find empirical probability. ≈ 0.373. Round to three decimal places. TRY IT YOURSELF 6 In Example 6, determine the probability that the next adult surveyed read only digital books during the last year. Answer: Page A38 EXAMPLE 6 Picturing the World It seems that no matter how unlikely an event is, somebody wants to know the probability that it will occur. The table below lists the probabilities that some rare events will happen. (Adapted from Life: The Odds by Gregory Arthur Baer) Event Probability Being audited by the IRS 0.6% Writing a New York Times best seller 0.005 Winning an Academy Award 0.00009 Having your identity stolen 0.5% Spotting a UFO 0.0000003 Which of these events is most likely to occur? Least likely? To explore this topic further, see Activity 3.1 on page 146. 3.1
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