An Example of Independence and Mutual Exclusiveness

From the literature, it is not clear if independence can be defined in cases when one of the events has probability zero. I've seen it given both ways.

If the independence condition is defined to be P(A|B)= P(A), then it's undefined when P(B) = 0. If the condition is P(AB) = P(A)*P(B), then that would apply to all events A and B, including those in which one of the probabilities is zero.


Assuming that the independence condition can be defined for any events A and B, here is an example of both independence and mutual exclusiveness.

Let the sample space be the unit interval, [0,1]. Let the probabiliity of an event be its Lebesgue measure.

Let A be the set of rationals in this unit interal and let B be the set of irrationals in the interval.

Then A and B are mutually exclusive and P(A) = 0 and P(B) = 1.

P(AB) = P(null set) = 0

P(A)*P(B) = 0*1 = 0.

Thus, P(AB) = P(A)*P(B) and A and B are independent as well as mutually exclusive.

The only question I have is whether the independence condition is undefined since P(A) = 0.

Jack
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