1. **Stating the problem:** We have 10 red balls and 5 black balls. Two balls are chosen randomly. We want to find the probability that the two balls chosen are of different colors. 2. **Formula and rules:** The probability of an event is given by: $$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$ 3. **Total number of ways to choose 2 balls from 15 balls:** $$\binom{15}{2} = \frac{15 \times 14}{2} = 105$$ 4. **Number of favorable outcomes (choosing 1 red and 1 black):** Number of ways to choose 1 red ball from 10: $$\binom{10}{1} = 10$$ Number of ways to choose 1 black ball from 5: $$\binom{5}{1} = 5$$ Total favorable outcomes: $$10 \times 5 = 50$$ 5. **Calculate the probability:** $$\text{Probability} = \frac{50}{105}$$ 6. **Simplify the fraction:** $$\frac{50}{105} = \frac{\cancel{5} \times 10}{\cancel{5} \times 21} = \frac{10}{21}$$ **Final answer:** The probability that the two balls chosen are of different colors is $$\boxed{\frac{10}{21}}$$