136 CHAPTER 3 Probability Using a Frequency Distribution to Find Probabilities A research organization is conducting a survey of randomly selected individuals to determine the ages of users of a social media application. So far, 3000 users of the application have been surveyed. The frequency distribution at the left shows the results. What is the probability that the next user surveyed is 25 to 34 years old? (Adapted from Statista) SOLUTION Because the responses are not equally likely to occur and are based on observations, use the formula for empirical probability. The event is a response of “25 to 34 years old.” The frequency of this event is 765. Because the total of the frequencies is 3000, the empirical probability that the next user is 25 to 34 years old is P1age 25 to 342 = 765 3000 = 0.255. TRY IT YOURSELF 7 Find the probability that the next user surveyed is 35 to 44 years old. Answer: Page A38 As you increase the number of times a probability experiment is repeated, the empirical probability (relative frequency) of an event approaches the theoretical probability of the event. This is known as the law of large numbers. As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event. Law of Large Numbers As an example of this law, suppose you want to determine the probability of tossing a head with a fair coin. You toss the coin 10 times and get 3 heads, so you obtain an empirical probability of 3 10. Because you tossed the coin only a few times, your empirical probability is not representative of the theoretical probability, which is 1 2. The law of large numbers tells you that the empirical probability after tossing the coin several thousand times will be very close to the theoretical or actual probability. The scatter plot below shows the results of simulating a coin toss 150 times. Notice that, as the number of tosses increases, the probability of tossing a head gets closer and closer to the theoretical probability of 0.5. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 30 60 90 120 150 Proportion that are heads Number of tosses Probability of Tossing a Head EXAMPLE 7 Ages Frequency, f 13 to 17 84 18 to 24 459 25 to 34 765 35 to 44 546 45 to 54 432 55 to 64 369 65 and over 345 Σf = 3000
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