Answer:
[tex]P(A_{1}|B ) =\frac{mp}{1+p(m-1)}[/tex]
Step-by-step explanation:
For mutually exclusive events as A1, A2, A3, etc, Bayes' theorem states:
[tex]P(A|B)= \frac{P(B|A)P(A)}{P(B)}[/tex]
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
[tex]P(A_{1})=p \\P(A_{2})=1-p \\P(B|A_{1})=1 \\P(B|A_{2})=\frac{1}{m} \\P(B)= P(A_{1})P(B|A_{1}) + P(A_{2})P(B|A_{2})= p+\frac{1-p}{m} \\[/tex]
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:
[tex]P(A_{1}|B)= \frac{P(B|A_{1})P(A_{1})}{P(B)}=\frac{(1)(p)}{p+\frac{1-p}{m}} =\frac{mp}{mp+1-p} =\frac{mp}{1+p(m-1)}[/tex]
