Question

Solve the following question and mark the best possible option.
In a test, an examinee either guesses or copies or knows the answers to a multiple-choice question with four choices. The probability that he makes 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.

Options

• `13/27`

• `24/29`

• `18/31`

• `17/31`

Answer 1

Let E₁ be the event that the answer is guessed E₂ be the event that the answer is copied, E₃ be the event that the examinee knows the answer and E be the event that the examinee answered correctly.
Given P(E₁) = `1/3, P(E_2) = 1/6`

Assume that events E₁, E₂ and E₃ are exhaustive.
P(E₁) + P(E₂) + P(E₃) = 1
P(E₃) = 1 - P(E₁) - P(E₂) = 1 - `1/3 - 1/6 = 1/2`

Now P(E/E₁) = Probability of getting a correct answer by guessing = `1/4`
[Since these are 4 alternatives]
P(E/E₂) = Probability of answering correctly by copying = `1/8`
P(E/E₃) = Probability of answering correctly by knowing = 1
Clearly (E₃/E) is the event he knew the answer to the question given that he correctly answered it.
∴ P(E₃/E) = `[P(E_3) P(E/E_3)]/[P(E_1) P(E/E_1) + P(E_2) P(E/E_2) + P(E_3) P(E/E_3)]`

∴ P(E₃/E) = `(1/2 xx 1)/(1/3 xx 1/4 + 1/6 xx 1/8 + 1/2 xx 1) = 24/29.`