With the throw of two dice, what is the probability that the sum will be a prime number?

Options

A. `4/11`
B. `7/18`
C. `5/12`
D. `5/11`

Explanation

To find the probability that the sum of the numbers on two thrown dice will be a prime number, we can follow these steps:

### Step 1: Identify the possible sums
When two dice are thrown, the smallest sum possible is 2 (1+1) and the largest sum possible is 12 (6+6). The possible sums when throwing two dice range from 2 to 12.

### Step 2: List the prime numbers between 2 and 12
Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The prime numbers between 2 and 12 are:
- 2
- 3
- 5
- 7
- 11

### Step 3: Count the total number of outcomes when throwing two dice
Each die has 6 faces, so the total number of outcomes when throwing two dice is:
\[ 6 \times 6 = 36 \]

### Step 4: Find the favorable outcomes for each prime sum
Now, we need to find how many combinations of the dice result in each of the prime sums listed above:

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

### Step 5: Calculate the total number of favorable outcomes
Now, we sum the number of favorable outcomes for all prime sums:
\[ 1 + 2 + 4 + 6 + 2 = 15 \]

### Step 6: Calculate the probability
The probability \( P(A) \) that the sum will be a prime number is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{15}{36} \]

### Step 7: Simplify the probability
We can simplify \( \frac{15}{36} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
\[ P(A) = \frac{15 \div 3}{36 \div 3} = \frac{5}{12} \]

### Final Answer
The probability that the sum will be a prime number when throwing two dice is:
\[ \frac{5}{12} \]

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