There are three categories of students in a class of 60 students: A: Very hardworking; B: Regular but not so hardworking; C: Careless and irregular 10 students are in category A, 30 in category B and rest in category C. It is found that the probability of students of category A, unable to get good marks in the final year examination is 0.002, of category B it is 0.02 and of category C, this probability is 0.20. A student selected at random was found to be one who could not get good marks in the examination. Find the probability that this student is of category C.
Given:
10 students are in category A
30 students are in category B
20 students are in category C
Let us assume U1, U2, U3 and A be the events as follows:
U1 = Choosing student from category A
U2 = choosing student from category B
U3 = choosing student from category C
A = Not getting good marks in final examination
Now,
⇒ 
⇒ 
⇒ 
⇒ P(A|U1) = P(student not getting good marks from category A)
⇒ 
⇒ P(A|U2) = P(student not getting good marks from category B)
⇒ 
⇒ P(A|U3) = P(student not getting good marks from category C)
⇒ 
Now we find
P(U3|A) = P(The student is from category C given that he didn’t get good marks in final examination)
Using Baye’s theorem:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
∴ The required probability is 
.