If A, B, and C events are independent, check if B and A \ C events...

If A, B, and C events are independent, check if B and A \ C events are independent or not.

Expert Solution

A , B , C are the events independent , we have to check if B and A|C are independent or not.

ANSWER :

YES , B and A|C are independent.

SHORT EXPLANATION :

Since , A, B , C are the independent events , so occurence of A have no effect given that the event C has already happened.

p( A|C ) = p ( A ) ................(1)

Now, Since , A, B are the independent events , p ( A B ) = p(A) * P(B) .....(2)

Now , B and A|C are independent if p( A|C B ) = p (A|C) * p(B)

But , p( A|C B ) = p ( A B ) ..........[ From (1) ] ................(3)

From (2) and (3)

p( A|C B ) = p ( A B ) = p (A|C) * p(B)

Therefore, p( A|C B ) = p (A|C) * p(B)

From the above proven statement we can safely conclude that A|C and B are independent.

orchestra answered 1 year ago

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