Conditional Probability and the Multiplication Rule 3.2 SECTION 3.2 Conditional Probability and the Multiplication Rule 147 Conditional Probability Independent and Dependent Events The Multiplication Rule What You Should Learn How to find the probability of an event given that another event has occurred How to distinguish between independent and dependent events How to use the Multiplication Rule to find the probability of two or more events occurring in sequence and to find conditional probabilities Have you ever ridden as a passenger in a self-driving vehicle? Age Yes No Total 18–64 202 549 751 65+ 23 196 219 Total 225 745 970 Sample Space Age Yes 18–64 202 65+ 23 Total 225 Conditional Probability In this section, you will learn how to find the probability that two events occur in sequence. Before you can find this probability, however, you must know how to find conditional probabilities. A conditional probability is the probability of an event occurring, given that another event has already occurred. The conditional probability of event B occurring, given that event A has occurred, is denoted by P1B0 A2 and is read as “probability of B, given A.” DEFINITION Finding Conditional Probabilities 1. Two cards are selected in sequence from a standard deck of 52 playing cards. Find the probability that the second card is a queen, given that the first card is a king. (Assume that the king is not replaced.) 2. The table at the left shows the results of a survey in which 970 U.S. adults were asked whether they have ever ridden as a passenger in a self-driving vehicle. Find the probability that an adult is 18 to 64 years old, given that the adult has ridden as a passenger in a self-driving vehicle. (Adapted from The Harris Poll) SOLUTION 1. Because the first card is a king and is not replaced, the remaining deck has 51 cards, 4 of which are queens. So, P1B A2 = 4 51 ≈ 0.078. The probability that the second card is a queen, given that the first card is a king, is about 0.078. 2. There are 225 adults who said they have ridden as a passenger in a self-driving vehicle. So, the sample space consists of these 225 adults, as shown at the left. Of these, 202 are 18 to 64 years old. So, P1B A2 = 202 225 ≈ 0.898. The probability that an adult is 18 to 64 years old, given that the adult has ridden as a passenger in a self-driving vehicle, is about 0.898. TRY IT YOURSELF 1 Refer to the survey in the second part of Example 1. Find the probability that an adult is 65 years old or older, given that the adult has not ridden as a passenger in a self-driving vehicle. Answer: Page A38 EXAMPLE 1
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