Problem
  • •A company is planning to market a new product. The company’s marketing vice-president is particularly concerned about the product’s superiority over the closest competitive product, which is sold by another company. The marketing vice-president assessed the probability of the new product’s superiority to be 0.7. This executive then ordered a market survey to determine the products superiority over the competition.
•
  • •The results of the survey indicated that the product was superior to its competitor.
•
  • •Assume the market survey has the following reliability:
  1. –If the product is really superior, the probability that the survey will indicate “superior” is 0.8.
  2. –If the product is really worse than the competitor, the probability that the survey will indicate “superior” is 0.3.
  • •After completion of the market survey, what should the vice-president’s revised probability assignment to the event “new product is superior to its competitors”?
 
P(Ai)
P(B|Ai)
P(Ai)* P(B|Ai)
Revised Probability
P(Ai|B)
A1
Probability product is superior
0.7
0.8
0.56
0.56/0.65 = 0.86
A2
Probability product is not superior
0.3
0.3
0.09
0.09/0.65 = 0.14
 
1.0
P(B) =
0.65
  

 

Why P(B) = 0.65?

Bayes Theorem

P (B п A1) = P(BıA1) x P(A1) = 0.8 x 0.7 = 0.56

P (B п A2) = P(BıA2) x P(A2) = 0.3 x 0.3 = 0.09

 

Complement of an event A ()

Definition:   The complement of an event A is the set of all outcomes in the sample space that are not included in the outcomes of event A. The complement of event A is represented by  (read as A bar).
 
Rule:   Given the probability of an event, the probability of its complement can be found by subtracting the given probability from 1.
  P() = 1 - P(A)

P(B) = P(B п A1) + P(B п 1)

       = P(B п A1) + P(B п A2)

       = 0.56 + 0.09

       = 0.65