Bayes Theorem is so important
in probability
theory that it has given rise
to a branch of statistics called Bayesian
statistics, as distinct from the more common
Frequentist approach.
The key difference is that the Bayesian
approach uses prior knowledge and belief
in estimating the probability of an outcome.
The Frequentist approach sets preconceptions
to one side and allows only the current
data to be considered.
Let A_{1}, A_{2},..........A_{k}
be a collection of mutually exclusive and
exhaustive possible events. Then for any
other possible event 'B':
The notation P(AB) means
the probability of 'A' given that 'B' has
occurred.
A medical procedure has a
90% probability of correctly diagnosing
a medical condition. However if the subject
does not have the condition there is a 1%
probability of making a faulty diagnosis.
If one person in ten thousand
suffers from the condition, what is the
probability that a person selected at random
who returns a positive result actually suffers
from the condition.
The probabilities are:


Probability 
B 
the probability of returning a positive
result 

A_{1} 
the prior (before the test) probability
of having the condition 
0.0001 
A_{2} 
the prior probability of not having
the condition (=1  A_{1}) 
0.9999 
P(BA_{1}) 
the probability of a positive result
if they have the condition 
0.90 
P(BA_{2}) 
the probability of a positive result
if they do not have the condition 
0.01 
P(A_{1}B) 
the probability of having the condition,
given a positive result 
0.0089 
Thus there is less than a 1% chance (0.89%)
that a person who returns a positive result
actually has the condition. This is highly
nonintuitive, until you realize that it
is because the condition is extremely rare,
and the number of false positives outweighs
the number of correct diagnoses. 