Probability of Independent Events If and are independent events the probability

# Probability of independent events if and are

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Probability of Independent Events If and are independent events, the probability that both and will occur is given by P A and B P A P B . B A B A Example 9 Probability of Independent Events A random number generator on a computer selects three integers from 1 to 20.What is the probability that all three numbers are less than or equal to 5? P A P A P A 1 4 1 4 1 4 1 64 . P A 5 20 1 4 . Example 10 Probability of Independent Events In 2004, approximately 65% of the population of the United States was 25 years old or older. In a survey, 10 people were chosen at random from the population. What is the probability that all 10 were 25 years old or older? (Source: U.S. Census Bureau) Solution Let represent choosing a person who was 25 years old or older. The probability of choosing a person who was 25 years old or older is 0.65, the probability of choosing a second person who was 25 years old or older is 0.65, and so on. Because these events are independent, you can conclude that the probability that all 10 people were 25 years old or older is Now try Exercise 53. P A 10 0.65 10 0.01. A Activities 1. You have two bags, each of which contains a red marble and a green marble. You select one marble from each bag. Write the sample space. Answer: 2. A bag contains four red, three yellow, and five green marbles. One marble is selected from the bag. What is the probability that it is a red marble? (Assume that the marbles are of the same size, shape, etc.) Answer: 3.A person draws two cards from a deck of 52 cards. What is the probability thatthe two cards drawn will both be facecards if the deck contains 12 face cards? 4. Use the student population of your class today. a. If two students are chosen at random, what is the probability that both will be female? b. If one student is chosen at random, what is the probability that this stu- dent was born in a month beginning with the letter J? Answers will vary depending on the class. 11 221 1 3 S RR , RG , GR , GG
644 Chapter 8 Sequences, Series, and Probability The Complement of an Event The complement of an event A is the collection of all outcomes in the sample space that are not in The complement of event is denoted by Because or and because and are mutually exclusive, it follows that So, the probability of is given by For instance, if the probability of winning a game is then the probability of losing the game is 3 4 . P A 1 1 4 P A 1 4 P A 1 P A . A P A P A 1. A A A 1 P A A . A A . Probability of a Complement Let be an event and let be its complement. If the probability of is then the probability of the complement is given by P A 1 P A . P A , A A A Exploration You are in a class with 22 other people. What is the probability that at least two out of the 23 people will have a birthday on the same day of the year? What if you know the probability of everyone having the same birthday? Do you think this information would help you to find the answer?