Problem Analysis:
We are given that a two-digit prime number, when 18 is added to it, gives another prime number with its digits reversed. We need to find how many such numbers are possible.

Key Points:
- A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
- In a two-digit number, the tens digit cannot be 0 because it would make the number a one-digit number. Therefore, the tens digit can only be 1, 2, 3, 4, 5, 6, 7, 8, or 9.
- The ones digit can be any digit from 0 to 9.

Solution:
To solve this problem, we can use a brute force approach and check each two-digit prime number to see if it satisfies the given condition.

Step 1: Find all the two-digit prime numbers.
- We can start by listing all the two-digit numbers from 10 to 99.
- For each number, we can check if it is prime by dividing it by all the numbers from 2 to the square root of the number. If it is divisible by any of these numbers, it is not prime.
- By applying this process, we can find all the two-digit prime numbers: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

Step 2: Check if adding 18 to each two-digit prime number gives another prime number with its digits reversed.
- For each two-digit prime number, we can add 18 to it and check if the resulting number is prime and has its digits reversed.
- For example, for the number 11, adding 18 gives 29, which is a prime number with its digits reversed.
- We can continue this process for all the two-digit prime numbers.

Step 3: Count the numbers that satisfy the given condition.
- By checking all the two-digit prime numbers, we find that there are two numbers that satisfy the given condition: 11 and 17.
- Therefore, the answer is 2.

Final Answer: The number of two-digit prime numbers that, when 18 is added, give another prime number with its digits reversed, is 2.