Question

In a test an examine either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he make a guess is 1/3 and the probability that he copies the answer is 1/6. The probability that his answer is correct given that he copied it, is 1/8. Find the probability that he knew the answer to the question given that he correctly answered it. 

Answer 1

Baye's theorem: E₁, E₂, E₃, . . . . . . . . . . . . . . ,E_n are mutually exclusive and exhaustive events and E is an event which takes place in conjunction with any one of E₁ then the probability of the event E₁ happening when the event E has taken place is given by

Let us define the events :

A₁ ≡ the examinee guesses the answer, 

A₂ ≡ the examinee copies the answer 

A₃ ≡ the examinee knows the answer, 

A ≡ the examinee answers correctly 

ATQ, P (A₁) = 1/3; P (A₂) = 1/6 

As any one happens out of A₁, A₂, A₃, these are mutually exclusive and exhaustive events. 

∴ P (A₁) + P (A₂) + P (A₃) = 1 

⇒ P (A₃) = 1 – 1/3 – 1/6 = 6 - 2 – 1/6 = 3/6 = 1/2 

Also we have, P(A| A1) = 1/4 

[∵ out of 4 choices only one is correct.] P (A| A²) = 1/8 

(given) P (A| A₃) = 1 

[If examinee knows the ans., it is correct. i.e. true event] 

To find P (A₃| A). By Baye's thm. P (A₃| A) 

= P (A₃|A) P (A₃)/P (A| A₁) P(A₁) + P (A| A₂) P(A₂) + P (A| A₃) P(A₃)

= (1(1/2))/(1/4 x 1/3 + 1/8 x 1/6 + (1 x 1)/2)

 = 1/2/29/48 = 1/2 x 18/29 = 24/29.