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2024 PEA Q18

In answering an MCQ with 44 choices, a student either knows the answer (with probability 14\tfrac14) or guesses (with probability 34\tfrac34). A guess is correct with probability 14\tfrac14. What is the probability that the student knew the answer, given that he answered correctly?

Reveal answer and solution

Answer

B

Solution

  1. 1

    Let KK = knows'' and $C$ = correct''. Then P(K)=14P(K)=\tfrac14, P(C∣K)=1P(C\mid K)=1, P(C∣Kc)=14P(C\mid K^{c})=\tfrac14. By total probability,

  2. 2
    P(C)=1⋅14+14⋅34=416+316=716. P(C)=1\cdot\tfrac{1}{4}+\tfrac{1}{4}\cdot\tfrac{3}{4}=\tfrac{4}{16}+\tfrac{3}{16}=\tfrac{7}{16}.
  3. 3

    By Bayes' theorem,

  4. 4
    P(K∣C)=P(C∣K)P(K)P(C)=1⋅14716=47. P(K\mid C)=\frac{P(C\mid K)P(K)}{P(C)}=\frac{1\cdot\tfrac14}{\tfrac{7}{16}}=\frac{4}{7}.

Answer structure / marking notes

No major trap beyond standard calculation care.

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Content note

Imported from public/resources/isi/msqe/solutions/pea/2024/ISI_MSQE_PEA_2024_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.