STAT 400 Lecture AL1 1 Examples for 1 5 Spring 2015 Dalpiaz In Neverland men constitute 60 of the labor force The rates of unemployment are 6 0 and 4 5 among males and females respectively A person is selected at random from Neverland s labor force a What is the probability that the person selected is a male and is unemployed b What is the probability that the person selected is a female and is unemployed Unemployed Male Female Total Employed Total c What is the probability that the person selected is unemployed Law of Total Probability P A P A B P A B P B P A B P B P A B In general m P A P B i P A B i i 1 d Suppose the person selected is unemployed What is the probability that a male was selected Bayes Theorem P B P A B P B A P B P A B P B P A B In general P B k P A B k P B k A m P B i P A B i i 1 k 1 m 2 In a presidential race in Neverland the incumbent Democrat D is running against a field of four Republicans R 1 R 2 R 3 R 4 seeking the nomination Political pundits estimate that the probabilities of R 1 R 2 R 3 and R 4 winning the nomination are 0 40 0 30 0 20 and 0 10 respectively Furthermore results from a variety of polls are suggesting that D would have a 55 chance of defeating R 1 in the general election a 70 chance of defeating R 2 a 60 chance of defeating R 3 and an 80 chance of defeating R 4 Assuming all these estimates to be accurate what are the chances that D will be a two term president 3 In Anytown 10 of the people leave their keys in the ignition of their cars Anytown s police records indicate that 4 2 of the cars with keys left in the ignition are stolen On the other hand only 0 2 of the cars without keys left in the ignition are stolen Suppose a car in Anytown is stolen What is the probability that the keys were left in the ignition 4 In a certain population the proportion of individuals who have a particular disease is 0 025 A test for the disease is positive in 94 of the people who have the disease and in 4 of the people who do not a Find the probability of receiving a positive reaction from this test b If a person received a positive reaction from this test what is the probability that he she has the disease c If a person received a negative reaction from this test what is the probability that he she doesn t have the disease
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