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Probability of Independent EventsIf and are independent events,the probability that both and willoccur is given byP Aand BP AP B.BABAExample 9Probability of Independent EventsA 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 AP AP A141414164.P A52014.Example 10Probability of Independent EventsIn 2004, approximately 65% of the population of the United States was 25 yearsold 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)SolutionLet represent choosing a person who was 25 years old or older. The probabilityof choosing a person who was 25 years old or older is 0.65, the probability ofchoosing 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 thatall 10 people were 25 years old or older isNow try Exercise 53.P A100.65100.01.AActivities1.You have two bags, each of whichcontains a red marble and a greenmarble. 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 isselected from the bag. What is theprobability 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 classtoday.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 beginningwith the letter J?Answers will vary depending on the class.1122113SRR, RG, GR, GG
644Chapter 8Sequences, Series, and ProbabilityThe Complement of an EventThe complement of an event Ais the collection of all outcomes in the samplespace that are notin The complement of event is denoted by Becauseor and because and are mutually exclusive, it follows thatSo, the probability of is given byFor instance, if the probability of winning a game isthen the probability of losingthe game is34.P A114P A14P A1P A.AP AP A1.AAA1P AA.AA.Probability of a ComplementLet be an event and let be its complement. If the probability of isthen the probability of the complement is given byP A1P A.P A,AAAExplorationYou are in a class with 22 otherpeople. What is the probabilitythat at least two out of the 23people will have a birthday onthe same day of the year? Whatif you know the probability ofeveryone having the samebirthday? Do you think thisinformation would help you tofind the answer?