To solve this problem, let's go through the steps carefully.

Step 1: Identify the First Nine Prime Numbers

The first nine prime numbers are: $2, 3, 5, 7, 11, 13, 17, 19, 23$

Step 2: Find All Possible Sums of Pairs

We need to select two different primes from these nine and determine their sums. The goal is to check if any of these pairs sum to 30.

Calculate the Pair Sums

Here are all possible pairs and their sums:

• $2 + 3 = 5$
• $2 + 5 = 7$
• $2 + 7 = 9$
• $2 + 11 = 13$
• $2 + 13 = 15$
• $2 + 17 = 19$
• $2 + 19 = 21$
• $2 + 23 = 25$
• $3 + 5 = 8$
• $3 + 7 = 10$
• $3 + 11 = 14$
• $3 + 13 = 16$
• $3 + 17 = 20$
• $3 + 19 = 22$
• $3 + 23 = 26$
• $5 + 7 = 12$
• $5 + 11 = 16$
• $5 + 13 = 18$
• $5 + 17 = 22$
• $5 + 19 = 24$
• $5 + 23 = 28$
• $7 + 11 = 18$
• $7 + 13 = 20$
• $7 + 17 = 24$
• $7 + 19 = 26$
• $7 + 23 = 30$ ⟵ This is a valid pair
• $11 + 13 = 24$
• $11 + 17 = 28$
• $11 + 19 = 30$ ⟵ This is another valid pair
• $11 + 23 = 34$
• $13 + 17 = 30$ ⟵ This is a third valid pair
• $13 + 19 = 32$
• $13 + 23 = 36$
• $17 + 19 = 36$
• $17 + 23 = 40$
• $19 + 23 = 42$

We found three pairs that sum to 30:

1. $7 + 23$
2. $11 + 19$
3. $13 + 17$

Step 3: Calculate the Probability

There are $9$ primes, and we are selecting 2 different ones, so the total number of ways to pick two primes from the nine is given by: $\binom{9}{2} = 36$

Since there are 3 pairs that result in a sum of 30, the probability is: $\frac{3}{36} = \frac{1}{12}$

Final Answer

The probability that the sum of two randomly selected primes from the first nine primes is 30 is: $\frac{1}{12}$

Do you want more details or have questions?

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1. What are other examples of problems involving probability and prime numbers?
2. How would the solution change if only odd primes were allowed?
3. What are other methods for calculating combinations in probability problems?
4. How does the concept of complement apply to probability problems?
5. Can the probability of a different target sum (e.g., 20) be calculated similarly?

Tip: For probability with combinations, list all possible pairs or outcomes clearly to avoid missing options.