SOLUTION: Show that two events A and B are independent if and only if P(A ∩ B) = P(A)P(B) when at least one of P(A) or P(B) is not zero. Thank you.

Algebra ->  Probability-and-statistics -> SOLUTION: Show that two events A and B are independent if and only if P(A ∩ B) = P(A)P(B) when at least one of P(A) or P(B) is not zero. Thank you.      Log On


   

Question 1177160: Show that two events A and B are independent if and only if P(A ∩ B) = P(A)P(B) when
at least one of P(A) or P(B) is not zero.
Thank you.
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
Many textbooks define independent events by that formula.  You can't prove
what is stated in a definition.

Two events A, B are said to be independent if and only if 

P(A ∩ B) = P(A)P(B).

All I can guess is that your textbook has defined it this way:

Two events A, B are said to be independent if and only if 

P(A|B) = P(A)

And the definition of conditional probability is 

P(A|B) = P(A ∩ B)/P(B).

If so, you can prove it by substituting P(A) for P(A|B)

P(A) = P(A ∩ B)/P(B)

and then multiplying both sides by P(B)

P(A)P(B) = P(A ∩ B)

Edwin