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# Bayes’s Theorem for Conditional Probability

• Difficulty Level : Easy
• Last Updated : 28 Jun, 2021

We strongly recommend to refer below post as a pre-requisite for this.

Conditional Probability

Bayes’s formula
Below is Bayes’s formula. The formula provides the relationship between P(A|B) and P(B|A). It is mainly derived from conditional probability formula discussed in the previous post.
Consider the below formulas for conditional probabilities P(A|B) and P(B|A) —-(1) —-(2)

Since P(B ∩ A) = P(A ∩ B), we can replace P(A ∩ B) in the first formula with P(B|A)P(A)

After replacing, we get the given formula.

Product Rule

Product rule states that

(1) So the joint probability that both X and Y will occur is equal to the product of two terms:

From the product rule : implies P(X|Y) = P(X)/P(Y) implies P(X|Y) = 1

Chain rule

When the above product rule is generalized we lead to chain rule. Let there are n events . Then, the joint probability is given by

(2) Bayes’ Theorem

From the product rule, and . As and are same .

(3) where .

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?