Question

for three persons A B C the chances of being selected as a manager of a firm are in the ratio of 4:1:2 respectively . The respective probability for them to introduce a radical change in marketing strategy are 0.3 0.8 0.5 .If the change does take place then find the probability that it is due to the appointment of B or C

Answer 1

P(A) = 4/7                 P(B) = 1/7           P(C) = 2/7

P(Introduction of Radical change by A)= P(R_A) = 0.3
P(introduction of Radical change by B) = P(R_B) = 0.8
P(introduction of radical change by C) = = P(R_C) = 0.5

Probability that a radical change is introduced in the marketing strategy
  P(R) = P(A) * P(R_A) + P(B) * P(R_B) + P(C) * P(R_C)
       = 3/7

Now we are given that a radical change R has been introduced.  Probability that it is introduced by A is:
      P( R_A | R)  = P(R_A Π R) / P(R)

   This is as per conditional probability  - Bayes theorem..

Intersection of the events  R_A and R  = P(A) * P(R_A) = 0.3 * 4/7 = 1.2/7

  So P(R_A | R) = (1.2/7) / (3/7) = 0.4

The complement of this probability is what is asked.

So  P[(R_B U R_C)  | R ] =  1  - 0.4  = 0.6

we can also compute the above answer by finding
P [ (R_B U R_C)  | R] = (1.8/7)  / (3/7) = 0.6