Answer :
Answer: [tex]\dfrac{12}{13}[/tex]
Step-by-step explanation:
Let A = he known the answer then A' = he guess the answer.
B = he answered it correctly
As per given , we have
[tex]P(A)=\dfrac{3}{4}\ \ ,\ \ P(A')=\dfrac{1}{4}[/tex]
[tex]P(B|A)=1[/tex]
[tex]P(B|A')=\dfrac{1}{4}[/tex]
By Bayes theorem , we have
[tex]P(A|B)=\dfrac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A')P(A')}\\\\ P(A|B)=\dfrac{1\times\dfrac{3}{4}}{1\times\dfrac{3}{4}+\dfrac{1}{4}\times\dfrac{1}{4}}\\\\= \dfrac{12}{13}[/tex]
The probability that the student knows the answer given that he answered it correctly is [tex]\dfrac{12}{13}[/tex] .