Did he know the answer?

Suppose there are 80 questions on the test. Based on the given information, he will know the answer to 40 questions (all correct), will guess on 20 questions (because there are 4 choices per question, he will get 5 of these correct), and he will copy 20 questions (he will get 15 of these correct). That makes a total of 40+5+15 = 60 questions correct, out of which he knew the answer to 40. Answer is 40/60 = 2/3.

$R$ = student is $R$ ight

$K$ = student $K$ nows the answer

Given that student either knows the answer (with success probability 1), guesses it (with success probability $\frac{1}{4}$ ) or copies it (with success probability $\frac{3}{4}$ )

$P(R) = \frac{1}{2} \times 1 + \frac{1}{4} \times \frac{1}{4} + \frac{1}{4} \times \frac{3}{4} = \frac{3}{4}$

$P(K) = \frac{1}{2}$

$P(R|K) = 1$

$P(K|R) = \frac{P(R|K) \times P(K)}{P(R)} = \frac{1 \times \frac{1}{2}}{\frac{3}{4}} = \frac{2}{3}$

The probability of him getting it right is (1/2)*1 + (1/4)(1/4) + (1/4)(3/4) = 12/16 = 3/4. Then the probability of him knowing the answer is (1/2)/(3/4) = 2/3, by sample spaces.