What is the probability that Natasha got it right?

Question

Billy answers on average 5 questions right out of 7. Natasha's average is 5 questions out of 9. They both attempt the same problem. If the question is answered correctly, what is the probability that Natasha got it right?

Answer 1

This problem requires the Bayes's Formula.

 

Let

P(B) = probability that Billy answers the question.

P(N) = probability that Natasha answers the question.

P(C|N) = 5/9

P(C|B) = 5/7

and

P(B|C) = probability that Billy got it right given the question is answered correctly.

P(N|C) = probability that Natasha got it right given the question is answered correctly.

 

then

P(B)  = P(N) = ½ {since both attempt the same problem}

P(C) = P(C|N)P(N) + P(C|B)P(B)

        = (5/9)(½) + (5/7)(½)

        = 40/63

 

So the probability that Natasha got it right if the question is answered correctly is

P(N|C) = P(C|N)P(N)/P(C)

           =(5/18)/(40/63)

           = 7/16

 

And the probability that Billy got it right if the question is answered correctly is

P(B|C) = 1 - P(N|C)

           = 9/16