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(D) Let $E_1$ be the event that the weak student knows the answer, and $E_2$ be the event that the weak student guesses the answer. Let $A$ be the eve

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$0.5$

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(D) Let $E_1$ be the event that the weak student knows the answer, and $E_2$ be the event that the weak student guesses the answer. Let $A$ be the event that the weak student gets the correct answer.We are given that the student knows $20 \%$ of the answers, so $P(E_1) = 0.20$. The probability of guessing is $P(E_2) = 1 - 0.20 = 0.80$.If the student knows the answer, the probability of getting it correct is $P(A|E_1) = 1$.Since there are $4$ options and only one is correct, the probability of guessing the correct answer is $P(A|E_2) = \frac{1}{4} = 0.25$.We need to find the probability that the student was guessing given that they got the answer correct, which is $P(E_2|A)$.Using Bayes' Theorem:$P(E_2|A) = \frac{P(E_2) \cdot P(A|E_2)}{P(E_1) \cdot P(A|E_1) + P(E_2) \cdot P(A|E_2)}$$P(E_2|A) = \frac{0.80 	imes 0.25}{(0.20 	imes 1) + (0.80 	imes 0.25)}$$P(E_2|A) = \frac{0.20}{0.20 + 0.20} = \frac{0.20}{0.40} = 0.5$.

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