Bayes’s formula for Conditional Probability

Conditional Probability

Example:
In a batch, there are 80% C programmers, and 40% are Java and C programmers. What is the probability that a C programmer is also Java programmer?

Let A --> Event that a student is Java programmer
    B --> Event that a student is C programmer
    P(A|B) = P(A ∩ B) / P(B)
           = (0.4) / (0.8)
           = 0.5
So there are 50% chances that student that knows C also 
knows Java 

Exercise:
1) What is the value of P(A|A)?

2) Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = 1/2, the values of P(A | B) and P(B | A) respectively are (GATE CS 2003)
(A) 1/4, 1/2
(B) 1/2, 1/14
(C) 1/2, 1
(D) 1, 1/2

Bayes’s formula for Conditional Probability

Example : Box P has 2 red balls and 3 blue balls and box Q has 3 red balls and 1 blue ball. A ball is selected as follows:

(i)  Select a box
(ii) Choose a ball from the selected box such that each ball in
     the box is equally likely to be chosen. The probabilities of
     selecting boxes P and Q are (1/3) and (2/3), respectively.  

Given that a ball selected in the above process is a red ball, the probability that it came from the box P is (GATE CS 2005)
(A) 4/19
(B) 5/19
(C) 2/9
(D) 19/30

Solution:

R --> Event that red ball is selected
B --> Event that blue ball is selected
P --> Event that box P is selected
Q --> Event that box Q is selected

We need to calculate P(P|R)?

P(R|P) = A red ball selected from box P
       = 2/5
P(P) = 1/3
P(R) = P(P)*P(R|P) + P(Q)*P(R|Q)
     = (1/3)*(2/5) + (2/3)*(3/4)
     = 2/15 + 1/2
     = 19/30

Putting above values in the Bayes's Formula
P(P|R) = (2/5)*(1/3) / (19/30)
       = 4/19

Exercise A company buys 70% of its computers from company X and 30% from company Y. Company X produces 1 faulty computer per 5 computers and company Y produces 1 faulty computer per 20 computers. A computer is found faulty what is the probability that it was bought from company X?