1. **Problem statement:** We have 10 balls numbered from 1 to 10. Two balls are drawn at random, and the sum of their numbers is 15. We want to find the probability that one of the balls has the number 7. 2. **Step 1: Identify all pairs of numbers from 1 to 10 whose sum is 15.** The pairs $(a,b)$ such that $a+b=15$ and $1 \leq a < b \leq 10$ are: $$ (5,10), (6,9), (7,8) $$ 3. **Step 2: Count the total number of pairs with sum 15.** There are 3 such pairs. 4. **Step 3: Count the pairs that include the number 7.** Only one pair includes 7: $(7,8)$. 5. **Step 4: Calculate the probability.** $$ \text{Probability} = \frac{\text{Number of favorable pairs}}{\text{Total pairs}} = \frac{1}{3} $$ 6. **Answer:** The probability that one of the balls has the number 7 given the sum is 15 is $\boxed{\frac{1}{3}}$.