Let us now discuss an interesting fact about
independence that should enhance our understanding.
Suppose that events A and B are independent.
Intuitively, if I tell you that A occurred, this does not
change your beliefs as to the likelihood that B will occur.
But in that case, this should not change your beliefs as to
the likelihood that B will not occur.
So A should be independent of B complement.
In other words, the occurrence of A tells you nothing about
B, and therefore tells you nothing about
B complement either.
This was an intuitive argument that if A and B are
independent, then A and B complement are also
independent.
But let us now verify this intuition
through a formal proof.
The formal proof goes as follows.
We have the two events, A and B. And event A can be broken
down into two pieces.
One piece is the intersection of A with B. So that's the
first piece.
And the second piece is the part of A which is outside B.
And that piece is A intersection with the
complement of B. So these are the two pieces that together
comprise event A.
Now, these two pieces are disjoint from each other.
And therefore, by the additivity axiom, the
probability of A is equal to the probability of A
intersection B plus the probability of A intersection
with B complement.
Using independence, the first term becomes probability of A
times probability of B. And we leave the second term as is.
Now let us move this term to the other side.
And we obtain that the probability of A intersection
with B complement is the probability of A minus the
probability of A times the probability of B. We factor
out the term probability of A, and we are left with 1 minus
probability of B. And then we recognize that 1 minus the
probability of B is the same as the probability of B
complement.
So we proved that the probability of A and B
complement occurring together is the product of their
individual probabilities.
And that's exactly the definition of A being
independent from B complement.
And this concludes the formal proof.