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Let us now discuss an
interesting fact about

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independence that should enhance
our understanding.

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Suppose that events A and
B are independent.

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Intuitively, if I tell you that
A occurred, this does not

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change your beliefs as to the
likelihood that B will occur.

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But in that case, this should
not change your beliefs as to

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the likelihood that
B will not occur.

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So A should be independent
of B complement.

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In other words, the occurrence
of A tells you nothing about

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B, and therefore tells
you nothing about

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B complement either.

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This was an intuitive argument
that if A and B are

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independent, then A and
B complement are also

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independent.

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But let us now verify
this intuition

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through a formal proof.

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The formal proof goes
as follows.

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We have the two events, A and
B. And event A can be broken

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down into two pieces.

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One piece is the intersection
of A with B. So that's the

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first piece.

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And the second piece is the part
of A which is outside B.

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And that piece is A intersection
with the

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complement of B. So these are
the two pieces that together

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comprise event A.

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Now, these two pieces are
disjoint from each other.

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And therefore, by the additivity
axiom, the

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probability of A is equal
to the probability of A

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intersection B plus the
probability of A intersection

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with B complement.

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Using independence, the first
term becomes probability of A

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times probability of B. And we
leave the second term as is.

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Now let us move this term
to the other side.

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And we obtain that the
probability of A intersection

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with B complement is the
probability of A minus the

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probability of A times the
probability of B. We factor

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out the term probability of A,
and we are left with 1 minus

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probability of B. And then we
recognize that 1 minus the

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probability of B is the same
as the probability of B

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complement.

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So we proved that the
probability of A and B

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complement occurring together
is the product of their

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individual probabilities.

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And that's exactly the
definition of A being

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independent from B complement.

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And this concludes
the formal proof.