Ex:            The following formulas define the behavior of conditional probabilities:

        (always true)

                                                       (if A and B independent)

                                                (if A and B independent)

For the following formulas, determine whether the formula is always true when A and B are independent.

a)     

b)     

c)      If and , then

d)      For an arbitrary event, C, A is independent of .

Sol'n:  a)  The first equation follows by direct application of the formulas for independent events:

             b) Because A and B are independent, we may immediately simplify the right-hand side of the equation:

Now we consider the left side of the equation:

Using the Law to Total Probability, we may relate to :

or

Again using the Law of Total Probability, we may relate to .

or

Substituting into our equation for , we have the following result:

or

The left side of the original equation now simplifies to :

Thus, the left and right sides of the original equations are equal whenever A and B are independent:

Note:       Our derivation shows that, when A and B are independent events, we also have independence of A and B ', A' and B, and A' and B '.

             c)  If and , then follows immediately. What is less immediately obvious is that this result implies that . In other words, the intersection of A and B is nonempty. Equivalently, A and B must overlap on a Venn diagram.

             d) For an arbitrary event, C, we investigate the independence of A and by examining the conditional probability for A.

Since may be any part of B, we may consider the case where :

Since it is not always true that P(A) = 1, the equation in (d) is not always true.