In an examination, a student can choose the order in which two questions ($\text{QuesA}$ and $\text{QuesB}$) 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. 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)?

Let:

• $P(A) = 0.8$, $\text{Marks}(A) = 10$

• $P(B) = 0.5$, $\text{Marks}(B) = 20$

Since answering the questions are independent events:

• $P(\text{both correct}) = P(A) \times P(B) = 0.8 \times 0.5 = 0.4$

### Case 1: First $\text{QuesA}$, then $\text{QuesB}$

1. $\text{QuesA}$ is wrong (probability $1 - 0.8 = 0.2$): Marks = $0$

2. $\text{QuesA}$ is correct, $\text{QuesB}$ is wrong (probability $0.8 \times (1 - 0.5) = 0.4$): Marks = $10$

3. Both $\text{QuesA}$ and $\text{QuesB}$ are correct (probability $0.8 \times 0.5 = 0.4$): Marks = $10 + 20 = 30$

$$\begin{aligned} E[\text{Marks}_{A \to B}] &= (0.2 \times 0) + (0.4 \times 10) + (0.4 \times 30) \\ &= 0 + 4 + 12 = 16 \end{aligned}$$

### Case 2: First $\text{QuesB}$, then $\text{QuesA}$

1. $\text{QuesB}$ is wrong (probability $1 - 0.5 = 0.5$): Marks = $0$

2. $\text{QuesB}$ is correct, $\text{QuesA}$ is wrong (probability $0.5 \times (1 - 0.8) = 0.1$): Marks = $20$

3. Both $\text{QuesB}$ and $\text{QuesA}$ are correct (probability $0.5 \times 0.8 = 0.4$): Marks = $20 + 10 = 30$

$$\begin{aligned} E[\text{Marks}_{B \to A}] &= (0.5 \times 0) + (0.1 \times 20) + (0.4 \times 30) \\ &= 0 + 2 + 12 = 14 \end{aligned}$$

Comparing the two expected values:

Therefore, the student should attempt First $\text{QuesA}$ and then $\text{QuesB}$, yielding an expected mark of $16$.