What is the probability that the sum of any two different single digit natural numbers is a prime number?

Options

A. `(5)/(27)`
B. `(7)/(18)`
C. `(1)/(3)`
D. None of these

Answer

To solve the problem, we need to find the probability that the sum of any two different single-digit natural numbers (from 1 to 9) results in a prime number.

### Step-by-Step Solution:

1. **Identify the Single-Digit Natural Numbers**:
The single-digit natural numbers are:
\[ 1, 2, 3, 4, 5, 6, 7, 8, 9 \]

2. **Calculate the Total Number of Outcomes**:
We need to find the total number of ways to choose 2 different numbers from these 9 numbers. This can be calculated using combinations:
\[ \text{Total Outcomes} = \binom{9}{2} = \frac{9 \times 8}{2} = 36 \]

3. **Find the Prime Sums**:
We will now find all pairs of different single-digit numbers whose sums are prime numbers. The prime numbers less than 18 (the maximum sum of two single-digit numbers) are:
\[ 2, 3, 5, 7, 11, 13, 17 \]

Now, we will check pairs of numbers:
- \(1 + 2 = 3\) (prime)
- \(1 + 4 = 5\) (prime)
- \(1 + 6 = 7\) (prime)
- \(1 + 10 = 11\) (not applicable)
- \(1 + 12 = 13\) (not applicable)
- \(1 + 16 = 17\) (not applicable)
- \(2 + 3 = 5\) (prime)
- \(2 + 5 = 7\) (prime)
- \(2 + 9 = 11\) (prime)
- \(3 + 4 = 7\) (prime)
- \(3 + 8 = 11\) (prime)
- \(4 + 7 = 11\) (prime)
- \(5 + 6 = 11\) (prime)
- \(4 + 9 = 13\) (prime)
- \(5 + 8 = 13\) (prime)
- \(6 + 7 = 13\) (prime)
- \(8 + 9 = 17\) (prime)

After checking all pairs, the valid pairs that yield prime sums are:
- (1, 2)
- (1, 4)
- (1, 6)
- (2, 3)
- (2, 5)
- (2, 9)
- (3, 4)
- (3, 8)
- (4, 7)
- (5, 6)
- (4, 9)
- (5, 8)
- (6, 7)
- (8, 9)

Counting these pairs, we find there are 14 pairs.

4. **Calculate the Probability**:
The probability \(P\) that the sum of two different single-digit natural numbers is a prime number is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{14}{36} \]

Simplifying this fraction:
\[ P = \frac{7}{18} \]

### Final Answer:
The probability that the sum of any two different single-digit natural numbers is a prime number is:
\[ \frac{7}{18} \]