UPSC NDA Math Model Paper 4

Let E1 be the event that ' Student knows the answer' , E2 be the event that ' Student guesses the answer and X be the event 'Student answered it correctly'.
Given: P(E1)=3∕4,P(E2)=1∕4 and probability that a student who guesses the answer will be correct is 1/4
Similarly, P(X|E1)= Probability student answered the question correctly given that he/she knows the answer =1P(X|E2)= Probability student answered the question correctly given that he/she guessed the answer = 1/4 ....... (Given)
Here, we have to find the value of P(E1|X)
As we know that according to bayes' theorem: P(Ei|A)=

P(Ei)×P(A|Ei)

n ∑ i=1 P(Ei)×P(A|Ei)

, i = 1, 2, …. , n
⇒P(E1|X) =

P(E1).P(X|E1)

[P(E1).P(X|E1)+P(E2).P(X|E2)]

⇒P(E1|X)=

3 4 .1

[ 3 4 .1+ 1 4 . 1 4 ]

=

12

13