 
Suppose
that being over 7 feet tall indicates with 60% probability that someone
is a basketball player, and carrying a basketball indicates this with 72%
probability. If you see someone who
is over 7 feet tall and carrying a basketball, what is the probability
that they're a basketball player?
If a and b are
the probabilities associated with two independent pieces of evidence,
then combined they indicate a probability of:
ab

ab + (1  a)(1  b)
So in this case our answer is:
(.60)(.72)

(.60)(.72) + (1  .60)(1  .72)
which is .794. There is a 79.4% chance that the person is a basketball player.
When there are more than two pieces of evidence, the formula expands
as you might expect:
abc

abc + (1  a)(1  b)(1  c)
In the case of spam filtering, what we want to calculate is the probability
that the mail is a spam, and the individual pieces of evidence
a, b, c, ... are the
spam probabilities associated with each of the words in the mail.
For a good explanation of the background, see:
http://www.mathpages.com/home/kmath267.htm.

