Suppose
is the event that a given patient has the disease, and
is the event of a positive test result.
We're given that
where
denotes the complement of an event
.
a. We want to find
. By the law of total probability, we have
That is, in order for
to occur, it must be the case that either
also occurs, or
does. Then from the definition of conditional probability we expand this as
so we get
b. We want to find
. Now, we can use Bayes' rule, but if you're like me and you find the formula a bit harder to remember, we can easily derive it.
By the definition of conditional probability,
We have the probabilities of
/
occurring given that
/
occurs, but not vice versa. However, we can expand the probability in the numerator to get a probability in terms of
being conditioned on
:
Meanwhile, the law of total probability lets us rewrite the denominator as
or in terms of conditional probabilities,
so that
which is exactly what Bayes' rule states. So we get
