$\frac{2}{3}$. If the student does not know the answer, then the student guesses the answer. Theprobability of the guessed answer being correct is $\frac{1}{4}$. Given that the student has answered
the question correctly, the probability that the student knows the correct answer is
Let event A: Student knows the correct answer,
event A’: Student guesses the answer,
event B: Answer is correct.
$\therefore P(A)=\frac{2}{3}, P\left(A^{\prime}\right)=\frac{1}{3}, P\left(B / A^{\prime}\right)=\frac{1}{4}$
Clearly, P(B/A) = 1 Required probability = P(A/B)
$\begin{aligned} & =\frac{ P ( A ) \cdot P ( B / A )}{ P ( A ) \cdot P ( B / A )+ P \left( A ^{\prime}\right) P \left( B / A ^{\prime}\right)} \\ & =\frac{\frac{2}{3} \times 1}{\frac{2}{3} \times 1+\frac{1}{3} \times \frac{1}{4}} \\ & =\frac{8}{9}\end{aligned}$
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