If A and B are independent events, prove that the following pairs of events are independent: A° and B, A and B° and A° and B.
What will be an ideal response?
To show that 
and B are independent, we show that 
= P(
)P(B). Now B = 
(A? B), and and (A ? B) are mutually exclusive. Therefore P(B) = + P(A ? B), from which it follows that = P(B)?P(A ? B). Since A and B are independent, P(A ? B) = P(A)P(B). Therefore 
= P(B) ? P(A)P(B) = P(B)[1 ? P(A)] = P(
)P(B). To show that A and 
are independent, it suffices to interchange A and B in the argument above. To show that and are independent, replace B with in the argument above, and use the fact that A and are independent.
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