In a test, an examinee either guesses or copies or knows that answer to a multiple-choice question which has four choices. The probability that he makes a guess is 1/3, and the probability that he copies is 1/6. The probability that his answer is correct, given he copied it, is 1/8, and the probability that his answer is correct, given he guessed it is 1/4. Find the probability that he knew the answer to the question, given that he answered it correctly.

We will be using Baye's Theorem to find the probability.

Answer: The probability that he knew the answer to the question, given that he answered it correctly is 24/29.

Here we have three mutually exclusive and exhaustive events.

Explanation:

P(g) = Probability of guessing = 1/3

P(c) = Probability of copying = 1/6

P(k) = Probability of knowing = 1 – 1/3 – 1/6 = 1/2

As the three events are mutually exclusive and exhaustive, we will be using Baye's theorem

P(a) = Probability that answer is correct

P(a|c) = Probability that his answer is correct, given he copied it = 1/8

P(a|g) = Probability that his answer is correct, given he guessed it = 1/4

According to Baye's theorem,

P(k|a)=(P(a|k).P(k))/(P(a|c)P(c)+P(a|k)P(k)+P(a|g)P(g))

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

= 24/29

Thus, the probability that he knew the answer to the question, given that he answered it correctly is 24/29.