In answering a question on a multiple choice test, a student either kn

1.	In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4 . What is the probability that the student knows the answer given that he answered it correctly?
Answer» Let E₁ : the event that the student knows the answer and E₂ : the event that the student guesses the answer. Then, E₁ and E₂ are mutually exclusive and exhaustive. Moreover, P(E₁) = 3/4 and P(E₂) = 1/4 Let E : the answer is correct. The probability that the student answered correctly, given that he knows the answer, is 1 i.e., P P(E/E₁) = 1 Probability that the students answered correctly, given that the he guessed, is 1/4 i.e., P(E/E₂) = 1/4 By using Baye’s theorem, we obtain P(E₁/E) = (P(E₁)P(E/E₁))/(P(E₁)P(E/E₁) + P(E₂)P(E/E₂)) = (3/4 x 1)/(3/4 x 1 + 1/4 x 1/4) = (3/4)/(3/4 + 1/16) = (12/16)/(12 + 1)/16 = 12/13

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4 . What is the probability that the student knows the answer given that he answered it correctly?

Answer»

Let E₁ : the event that the student knows the answer and E₂ : the event that the student guesses the answer.

Then, E₁ and E₂ are mutually exclusive and exhaustive. Moreover,

P(E₁) = 3/4 and P(E₂) = 1/4

Let E : the answer is correct. The probability that the student answered correctly, given that he knows the answer, is 1 i.e., P P(E/E₁) = 1

Probability that the students answered correctly, given that the he guessed, is 1/4

i.e., P(E/E₂) = 1/4

By using Baye’s theorem, we obtain

P(E₁/E) = (P(E₁)P(E/E₁))/(P(E₁)P(E/E₁) + P(E₂)P(E/E₂))

= (3/4 x 1)/(3/4 x 1 + 1/4 x 1/4) = (3/4)/(3/4 + 1/16) = (12/16)/(12 + 1)/16 = 12/13