2 Suppose that PA 05PB 04PC 015 PA U B 08PA U C 065PB U

2. Suppose that P(A) = 0.5,P(B) = 0.4,P(C) = 0.15, P(A U B) = 0.8,P(A U C) = 0.65,P(B U C) = 0.55. a. Compute P(A^?). b. Compute P(A B), and use the result to determine if A and B are mutually exclusive. c. Determine if A and C are mutually exclusive. Explain briefly. d. Describe in simple words what (A U B U C)^? represents. Determine P[(A U B U C)^?].

Solution

a)

P(A\') = 1 - P(A) = 1 - 0.5

P(A\') = 0.5 [ANSWER]

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B)

P(A and B) = P(A) + P(B) - P(A U B) = 0.5 + 0.4 - 0.8

P(A and B) = 0.1 [ANSWER]

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C)

If A and C are mutually exclusive, P(A and C) = 0.

As

P(A and C) = P(A) + P(C) - P(A U C) = 0.5 + 0.15 - 0.65

P(A and C) = 0

THUS, THEY ARE MUTUALLY EXCLUSIVE! [ANSWER]

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d)

(A U B U C)\' is the set that includes all elements outside the sets A, B, and C.

In this case, as C is mutually exclusive from A and B,

P(A U B U C) = P(A U B) + P(C) = 0.8 + 0.15 = 0.95

Thus,

P(A U B U C)\' = 1 - P(A U B U C) = 1 - 0.95

P(A U B U C)\' = 0.05 [ANSWER]