Two dice are tossed. What is the probability that the total score is a prime number.

1. Analyze the question

We are tossing two fair dice. We need to find the probability that the sum of the numbers showing on the two dice is a prime number.

2. Explain the concept

• Sample Space: When two dice are tossed, each die has 6 faces. The total number of possible outcomes is $6 \times 6 = 36$.
• Prime Numbers: A prime number is a number greater than 1 that has only two factors: 1 and itself. The possible sums when two dice are tossed range from 2 ($1+1$) to 12 ($6+6$). The prime numbers in this range are 2, 3, 5, 7, and 11.
• Probability Formula: $P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

3. Show the solution

Let's list the combinations that result in each prime sum:

• Sum = 2: (1, 1) — 1 outcome
• Sum = 3: (1, 2), (2, 1) — 2 outcomes
• Sum = 5: (1, 4), (4, 1), (2, 3), (3, 2) — 4 outcomes
• Sum = 7: (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3) — 6 outcomes
• Sum = 11: (5, 6), (6, 5) — 2 outcomes

Now, add the number of favorable outcomes: $1 + 2 + 4 + 6 + 2 = 15$

Calculate the probability: $P(\text{Prime}) = \frac{15}{36}$

Simplify the fraction by dividing both the numerator and the denominator by 3: $\frac{15 \div 3}{36 \div 3} = \frac{5}{12}$

4. Identify the answer

The calculated probability is $\frac{5}{12}$, which matches option A.

5. Tip

When dealing with two dice, remember the "sum distribution" pattern: the number of ways to get a sum increases as you move toward 7 (the most frequent sum) and decreases as you move toward 2 or 12. This can help you quickly verify if your count is reasonable!