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.
