Bayes Conditional Probability

One of the most interesting applications of the results of the probability theory includes estimating an unknown probability and making decisions on the basis of new information. Since from World War II, a considerable bossy of knowledge has developed termed as Bayesian decision theory whose aim is the solution of problems involving decision-making under uncertainty.

The concept of conditional probability takes into account when information about the occurrence of one event is used to predict the probability of another event. This concept can be prolonged to ''revise'' probabilities based on new information and to determine the probability that a particular effect was due to a specific cause. The process for reviving these probabilities is termed as Bayes theorem.

The Bayes theorem is named after the British mathematician Thomas Bayes (1702-61) and published in 1763 in a short paper has become one of the most famous in the history of science and one of the most controversial. His contribution consists mainly of a unique technique for calculating the conditional probability. For e.g., a sample output of 2 defectives in 50 trials (event A) might be used to estimate the probability that a machine is not working correctly (event B) or your might use the results of your first examination in statistics as  sample evidence in estimating the probability of getting a first class (event B).

The Bayes' theorem is based on the formula for conditional probability explained earlier.

A₁ and A₂ = the set of event which are mutually exclusive (the two events cannot occur together) & exhaustive (the combination of the two event is the whole experiment); and

B = A simple event which intersects each of the A events

The part of B which is within A₁ represents the ''A₁ and B'' and the part of B within A₂ represents the area ''A₂ and B'',

Then the probability of event A₁ given event B is
                                        
P (A₁/B) = P (A and B)
                      P (B)

And, similarly the probability of event A₂, given B, is
                               
P (A₂/B) = p (A₂ and B)
                      P (B)

Where, P (B) = p (A₁ and B) + p (A₂ and B).
                          
P (A₁ and B) = P (A₁) x p (B/A₁). And
                       
P (A₂ and B) = p (A₂) x P (B/ A₂)

In normal, let A₁, A₂, A₃ ...................A_i......................., A_n be a set of n mutually exclusive & collectively exhaustive events. If B is another event such that P (B) is not zero, then
                       
P (A₁/B) = p (B/A) P (A1)/ Σ p (B/A₁) p (A₁)
                      I=1

Illustration: -

suppose that a factory has two machines. The Past records show that machine 1 produces 30% of the items of output & machine 2 produces 70% of the items. Further, 5% of the items produced by machine 1 were defective and only 1% produced by machine 2 were defective. If a defective item is drawn at random, what is the probability that the defective item was produced by machine 1 or machine 2?

Solution: - let A₁ = the event of drawing an item produced by machine 1,
                      
A₂ = the event of drawing an item produced by machine 2
   
and, B = the event of drawing a defective item produced either by machine 1 or machine2,
 then from the first information.
                     
P A (1) = 30% = 0.30, P (A₂) = 70% = 0.70

from the additional information.
                    
P (B/A₁) = 5% = 0.05, P (B/ A₂) = 1% = 0.1

the required values are tabulated below:
                                     
Computation of posterior probabilities