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

A 
they have the condition 

B 
they return a positive result 

A_{1} 
they have the condition 
0.0001 
A_{2} 
they do not have the condition 
0.9999 
P(BA_{1}) 
they return a positive result given that they have the condition 
0.90 
P(BA_{2}) 
they return a positive result when they do not have the condition 
0.01 
P(A_{1}B) 
the probability they have the condition, given the positive result 
0.0089 
This is highly nonintuitive. The answer stands because the condition is extremely rare, and the number of false positives outweighs the number of correct diagnoses.
Complementary Probability 

The complement of a set A is the set of all outcomes that are not contained in A, denoted by A':
The outcome of all the events that are in both A and B. Intersection is roughly equivalent to "and":
The multiplication rule states that:
There is a 0.01 (1%) probability that a seal will fail if the joint is not coated with a sealing compound before assembly. The proportion of joints that are not coated is 0.05 (5%).
What is the probability that a seal will fail.
P(AB) 
Probability that the seal will fail if not coated 
0.01 
P(B) 
Probability the seal will not be coated 
0.05 

0.01 x 0.05 
=0.0005 
Events that are mutually exclusive, eg. a head and a tail on a coin toss:
The chance of something happening. The probability of getting a head on a coin toss is 0.5.
Probability can also be viewed as the fraction of occurrences over a large number of trials. If the results from many coin tosses are recorded the proportion of heads will be close to 50% (assuming a fair game).
The outcome of all the events that include A or B. Union is roughly equivalent to "or":
A diagram used to illustrate the concepts of probability. Venn diagrams are used in the entries for 'Union', 'Intersection' and 'Complement'.
