1. Total possible outcomes

When two fair dice are thrown, the total number of equally likely outcomes is: $36$

2. Find the outcomes for each case

(i) Doublet

A doublet means both dice show the same number: $(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)$ So, number of doublets = $6$

(ii) Sum of 9

The outcomes giving sum $9$ are: $(3,6),(4,5),(5,4),(6,3)$ So, number of outcomes = $4$

Note that none of these is a doublet, since no doublet has sum $9$.

(iii) Any other outcome

Remaining outcomes: $36-6-4=26$

3. Probabilities

• Probability of doublet: $P(\text{doublet})=\frac{6}{36}=\frac{1}{6}$

• Probability of sum $9$: $P(\text{sum }9)=\frac{4}{36}=\frac{1}{9}$

• Probability of any other outcome: $P(\text{other})=\frac{26}{36}=\frac{13}{18}$

4. Expected value

Let the expected gain be $E$.

Then $E=15\left(\frac{1}{6}\right)+12\left(\frac{1}{9}\right)-6\left(\frac{13}{18}\right)$

Now compute each term: $15\cdot \frac{1}{6}=\frac{15}{6}=\frac{5}{2}$ $12\cdot \frac{1}{9}=\frac{12}{9}=\frac{4}{3}$ $6\cdot \frac{13}{18}=\frac{13}{3}$

So, $E=\frac{5}{2}+\frac{4}{3}-\frac{13}{3}$ $E=\frac{5}{2}-3$ $E=\frac{5-6}{2}=-\frac{1}{2}$

5. Interpretation

Since the expected value is negative, the person has an expected loss of: $\frac{1}{2} \text{ rupee}$

6. Option check

• A: $\frac{1}{4}$ loss $\to$ incorrect
• B: $\frac{1}{2}$ gain $\to$ incorrect
• C: $\frac{1}{2}$ loss $\to$ correct
• D: $2$ gain $\to$ incorrect