... p(Q|P) is the proportion of things that have property Q and property P
within all things that have P; i.e.,
the proportion of women with breast cancer and a positive mammography within
the group of all women with positive mammographies.
If there are
641 women with breast cancer and positive mammographies,
7915 women with positive mammographies, and
89,031 women,
then p(Q&P) is the probability of getting one of those 641 women
if you're picking at random from the entire group of 89,031,
while p(Q|P) is the probability of getting one of those 641 women
if you're picking at random from the smaller group of 7915.
Here's another quote :
In a sense, p(Q|P)really means p(Q&P|P), but specifying the extra P all the
time would be redundant. You already know it has property P, so the property
you're investigating is Q - even though you're looking at the size of group
Q&P within group P, not the size of group Q within group P (which would be
nonsense).
This is what it means to take the property on the right-hand side as given;
it means you know you're working only within the group of things that have
property P. When you constrict your focus of attention to see only this
smaller group, many other probabilities change. If you're taking P as given,
then p(Q&P) equals just p(Q) - at least, relative to the group P.
The old p(Q), the frequency of
"things that have property Q within the entire sample",
is revised to the new frequency of
"things that have property Q within the subsample of things that have property P".
If P is given, if P is our entire world, then looking for Q&P
is the same as looking for just Q.
