Two dice are thrown simultaneously. The probability that the sum of the numbers appearing on them is a prime number is

Options

A. `5/12`
B. `1/2`
C. `7/12`
D. `2/3`

Answer

To solve the problem of finding the probability that the sum of the numbers appearing on two dice is a prime number, we will follow these steps:

### Step 1: Determine the total number of outcomes
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is:

\[ \text{Total outcomes} = 6 \times 6 = 36 \]

### Step 2: Identify the prime numbers between 1 and 12
The possible sums when rolling two dice range from 2 (1+1) to 12 (6+6). We need to identify the prime numbers within this range. The prime numbers between 1 and 12 are:

\[ 2, 3, 5, 7, 11 \]

### Step 3: Calculate the pairs that yield prime sums
Now we will find the pairs of dice rolls that result in each of these prime sums:

- **Sum = 2**: (1, 1) → 1 pair
- **Sum = 3**: (1, 2), (2, 1) → 2 pairs
- **Sum = 5**: (1, 4), (2, 3), (3, 2), (4, 1) → 4 pairs
- **Sum = 7**: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) → 6 pairs
- **Sum = 11**: (5, 6), (6, 5) → 2 pairs

Now, let's count the total number of pairs that result in a prime sum:

\[ 1 + 2 + 4 + 6 + 2 = 15 \text{ pairs} \]

### Step 4: Calculate the probability
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of outcomes. Therefore, the probability \( P \) that the sum of the numbers on the two dice is a prime number is:

\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{15}{36} \]

### Step 5: Simplify the probability
Now we simplify \( \frac{15}{36} \):

\[ \frac{15}{36} = \frac{5}{12} \]

### Conclusion
Thus, the probability that the sum of the numbers appearing on the two dice is a prime number is:

\[ \frac{5}{12} \]