STAT 400 Lecture AL1 Examples for 1.5 Spring 2015 Dalpiaz 1. 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 Employed Total Male Female 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, P( A ) = ∑=mi1P( B i ) ⋅ P( A B i ) 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 ) = , k = 1, … , m. ∑=mi1P( B i ) ⋅ P( A B i )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
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