Suppose that if a person is randomly chosen in Los Angeles, there is a 1/3 proba

Question

Suppose that if a person is randomly chosen in Los Angeles, there is a 1/3 probability that he / she is
a Dodgers fan, a 1/4 probability that he / she is a Lakers fan, and a 1/2 probability that he / she is
either a Dodgers fan or a Lakers fan.
(a) What is the probability that Emma is not a Dodgers fan, or not a Lakers fan?
(b) What is the probability that Mia is a Dodgers fan, but not a Lakers fan?
(c) Given that Ryan is a Lakers fan, what is the probability that he is also a Dodgers fan?
(d) Given that Sebastian is not a Lakers fan, what is the probability that she is a Dodgers fan?
(e) Are the events John is a Dodgers fang and John is a Lakers fan independent?
(f) Are the events Keith is a Dodgers fan and Keith is a Lakers fan mutually exclusive?

Explanation / Answer

Let A denote Didgers fan and B denote Lakers fan.

P(A) = 1/3 P(B) = 1/4 P(A U B) = 1/2

P(A B) = P(A) + P(B) - P(A U B) = 1/3 + 1/4 - 1/2 = 1/12

(a) P(A' U B') = P(A B)' = 1 - 1/12 = 11/12.

(b) P(A - B) = P(A B') = P(A) - P(A B) = 1/3 - 1/12 = 1/4.

(c) P(A|B) = P(A B) / P(B) = 1/12 / 1/4 = 1/3.

(d) P(B'|A) = P(A B') / P(A) = 1/4 / 1/3 = 3/4.

(e) Since P(A|B) = P(A), the events are independent.

(f) Since P(A B) is not zero, A and B are not mutually exclusive.

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