In an examination, a student can choose the order in which two questions (QuesA and QuesB) must be attempted.
- If the first question is answered wrong, the student gets zero marks.
- If the first question is answered correctly and the second question is not answered correctly, the student gets the marks only for the first question.
- If both the questions are answered correctly, the student gets the sum of the marks of the two questions.
The following table shows the probability of correctly answering a question and the marks of the question respectively.
\[
\begin{array}{c|c|c}
\text{Question} & \text{Probability of answering correctly} & \text{Marks}
\hline
\text{QuesA} & 0.8 & 10
\text{QuesB} & 0.5 & 20
\end{array}
\]
Assuming that the student always wants to maximize her expected marks in the examination, in which order should she attempt the questions and what is the expected marks for that order (assume that the questions are independent)?
- If the first question is answered wrong, the student gets zero marks.
- If the first question is answered correctly and the second question is not answered correctly, the student gets the marks only for the first question.
- If both the questions are answered correctly, the student gets the sum of the marks of the two questions.
The following table shows the probability of correctly answering a question and the marks of the question respectively.
\[ \begin{array}{c|c|c} \text{Question} & \text{Probability of answering correctly} & \text{Marks}
\hline \text{QuesA} & 0.8 & 10
\text{QuesB} & 0.5 & 20
\end{array} \] Assuming that the student always wants to maximize her expected marks in the examination, in which order should she attempt the questions and what is the expected marks for that order (assume that the questions are independent)?
Show Hint
- First QuesA and then QuesB. Expected marks 14.
- First QuesB and then QuesA. Expected marks 14.
- First QuesB and then QuesA. Expected marks 22.
- First QuesA and then QuesB. Expected marks 16.
The Correct Option is D
Solution and Explanation
Step 1: Expected marks if QuesA is attempted first.
- Probability QuesA correct = \(0.8\). If wrong, marks \(=0\).
- If QuesA is correct: marks \(=10\), and then attempt QuesB.
- Expected additional marks from QuesB \(= 0.5 \times 20 = 10\).
So, expected marks:
\[
0.8 \times (10 + 10) = 0.8 \times 20 = 16
\]
Step 2: Expected marks if QuesB is attempted first.
- Probability QuesB correct = \(0.5\). If wrong, marks \(=0\).
- If QuesB is correct: marks \(=20\), then attempt QuesA.
- Expected additional marks from QuesA \(= 0.8 \times 10 = 8\).
So, expected marks:
\[
0.5 \times (20 + 8) = 0.5 \times 28 = 14
\]
Step 3: Comparison.
\[
\text{Expected marks (A first)} = 16 > \text{Expected marks (B first)} = 14
\]
Step 4: Conclusion.
To maximize expected marks, the student should attempt QuesA first and then QuesB.
Final Answer: (D)
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