Let A B events with PA 05 and PB 08 a Are A and B disjoint

Let A, B events with P(A) = 0.5 and P(B) = 0.8. (a) Are A and B disjoint? (b) Suppose further that P(A B) = 0.9. Are A and B independent?

Solution

A)

If A and B are disjoint, then

P(A) + P(B) <= 1

However,

P(A) + P(B) = 0.5 + 0.8 = 1.3 > 1.

Thus, THEY ARE NOT DISJOINT.

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B)

If A and B are independent,

P(A|B) = P(A)

As

P(A|B) = P(A and B) / P(B)

and

P(A and B) = P(A) + P(B) - P(A U B) = 0.5 + 0.8 - 0.9 = 0.4

Thus,

P(A|B) = 0.4/0.8 = 0.5

Thus,

P(A|B) = P(A) = 0.5 [THEY ARE INDEPENDENT!]