Question

In answering on a multiple choice test, a student either know the answer or guesses. Let p be the probability that the students knows the answer and 1-p be the probability that the student guesses. Assume that a student who guesses at the answer will be correct with probability 1/m, where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer to a question when he or she answered it correctly?

Answer 1

`P(A_(1)|B ) =(mp)/(1+p(m-1))`

Explanation:

For mutually exclusive events as A1, A2, A3, etc, Bayes' theorem states:

`P(A|B)= (P(B|A)P(A))/(P(B))`

P(A|B) is a conditional probability: the likelihood of event A occurring given that B is true.

P(B|A) is a conditional probability: the likelihood of event B occurring given that A is true.

P(A) is the probability that A occurs

P(B) is the probability that B occurs

For this problem:

A1 is the probability that the student knows the answer

A2 is the probability that the student guesses the answer

B is the probability that the student answer correctly

`P(A_(1))=p \\P(A_(2))=1-p \\P(B|A_(1))=1 \\P(B|A_(2))=(1)/(m) \\P(B)= P(A_(1))P(B|A_(1)) + P(A_(2))P(B|A_(2))= p+(1-p)/(m) \\`

P(B|A₁) means the probability that the answer is correct when he knew the answer

P(B|A₂) means the probability that the answer is correct when he guessed the answer

P(A₁|B) means the probability that he knew the answer when the answer was correct

Replacing everything in the Bayes' theorem you get:

`P(A_(1)|B)= (P(B|A_(1))P(A_(1)))/(P(B))=((1)(p))/(p+(1-p)/(m)) =(mp)/(mp+1-p) =(mp)/(1+p(m-1))`