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Probabilities of Complementary Events
Remember that if A is some event, and
Ac is the complement of A in the
sample space
,
then A and
Ac are disjoint (mutually exclusive,
have nothing in common). Also, their union is the entire sample space:
AAc
=
.
Since
is the same set as
AAc, it has to have the same probability.
Moreover, we know that probability has to be 1:
1=P()
=
P(AAc).
But that is a disjoint union, and the probability of the union is the sum
of the probabilities:
1=P(A)+P(Ac).
So what this amounts to is saying that the probability that something
happens, plus the probability that it doesn't happen, is one. So for
instance if
you know 20% of your sample are smokers, then you know 80% have to be
nonsmokers.
Probability of Empty Event is zero
Since the empty set (impossible event) has nothing in common with
anything, we can write the sample space as the union of the
sample space with the empty set. Then the probability of the disjoint
union is the sum of the probabilies, so
P(
)
=
P(
)
=
P(
)
+
P(
)
So we get 1=1+P(
)
. Subtract off one from either side, and we learn that the probability
of something impossible has to be zero.
.
)
=
P(