1. **State the problem:** Priya flips two unbiased coins and rolls a fair five-sided die labeled 1 to 5. The final score $X$ is the number of heads plus the number on the die. We need to find the probability that $X=2$. 2. **Understand the variables:** - Number of heads from two coins can be 0, 1, or 2. - Die roll can be 1, 2, 3, 4, or 5. 3. **Express the event $X=2$:** $$X = \text{number of heads} + \text{die roll} = 2$$ 4. **Find all pairs (heads, die) such that heads + die = 2:** - If heads = 0, die = 2 - If heads = 1, die = 1 - If heads = 2, die = 0 (not possible since die minimum is 1) So possible pairs: (0,2) and (1,1). 5. **Calculate probabilities:** - Probability of 0 heads in 2 coin flips: $$P(0) = \binom{2}{0} \left(\frac{1}{2}\right)^0 \left(\frac{1}{2}\right)^2 = 1 \times 1 \times \frac{1}{4} = \frac{1}{4}$$ - Probability of 1 head in 2 coin flips: $$P(1) = \binom{2}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^1 = 2 \times \frac{1}{2} \times \frac{1}{2} = \frac{2}{4} = \frac{1}{2}$$ - Probability of die roll = 1 or 2 (each is $\frac{1}{5}$ since die is fair) 6. **Calculate combined probabilities for each pair:** - For (0 heads, die=2): $$P = P(0) \times P(2) = \frac{1}{4} \times \frac{1}{5} = \frac{1}{20}$$ - For (1 head, die=1): $$P = P(1) \times P(1) = \frac{1}{2} \times \frac{1}{5} = \frac{1}{10}$$ 7. **Add probabilities for all favorable outcomes:** $$P(X=2) = \frac{1}{20} + \frac{1}{10} = \frac{1}{20} + \frac{2}{20} = \frac{3}{20}$$ **Final answer:** $$\boxed{\frac{3}{20}}$$