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(10 pts) Gaydar
Some people use “gaydar” to determine if someone is gay/lesbian. Use the data below (from this journal article) to answer the following questions. Assume 5% of people are actually gay/lesbian. Gaydar correctly spots picks a gay/lesbian person 56% of the time. Gaydar has a false positive rate (incorrectly identifying a heterosexual as a homosexual) of 30%..
(3 pts) Suppose you look at someone and decide he is gay using your gaydar. What is the probability that you are right? Use Bayes’ rule to find out.
(2 pts) In San Francisco, the city with highest LGB population by percentage, 15.4% of the city population is LGB. How does this affect the probability that your gaydar is correct? Use natural frequencies.
(3 pts) Both you and your friend think someone looks gay. What is the probability that you’re correct? Assume your assessments are independent. (You can also assume conditional independence, i.e. that your assessments are independent even when conditioned on the event that the person is gay. In general, independence does not imply conditional independence.)
(2 pts) Is the assumption of independence in the previous question reasonable? Why/why not?
(4 pts) Check the following two statements. Are they correct or incorrect? Show your reasoning.
(2 pts) \( P(A^\complement | B) + P(A|B)=1 \)
(2 pts) \( P(A|B^\complement) + P(A|B)=1 \)
(7 pts) Given that A, B, and C are mutually independent, prove the following:
(3 pts) \( A^\complement \) and \( B^\complement \) are independent.
(1 pt) \( A^\complement \) and \( C^\complement \) are independent, and \( B^\complement \) and \( C^\complement \) are independent. (Think about how to do this as efficiently as possible.)
(3 pts) \( A^\complement \), \( B^\complement \), and \( C^\complement \) are mutually independent.