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Let P(A) = 0.4 and P(A u B) = 0.6:
(a) For what value of P(B) are A and B mutually exclusive?
(b) For what value of P(B) are A and B independent?
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Part (a) solution
Events A and B are mutually exclusive if and only if P(A U B) = P(A) + P(B).
In our case, it should be
0.6 = 0.4 + P(B),
which gives P(B) = 0.6 - 0.4 = 0.2.
ANSWER. P(B) = 0.2.
Part (b) solution
We have a general formula P(A U B) = P(A) + P(B) - P(A n B), which is valid for all events A and B.
If A and B are independent, then P(A n B) = P(A)*P(B).
So, in our case should be
0.6 = 0.4 + P(B) - 0.4*P(B),
which implies
0.6 - 0.4 = 0.6*P(B)
0.2 = 0.6*P(B)
P(B) = = .
ANSWER. P(B) should be .
Solved.