To solve this problem, let's break it down:

We have a total of 20 balls: 10 red and 10 white. We are interested in the probability of drawing 3 balls of the same color (either 3 red balls or 3 white balls).

Total number of possible outcomes:

When drawing 3 balls, the total number of ways to choose 3 balls out of the 20 available is calculated using combinations (denoted as $\binom{n}{k}$, where $n$ is the total number of items and $k$ is the number of items to choose):

$\text{Total possible outcomes} = \binom{20}{3}$

Favorable outcomes:

1. Drawing 3 red balls: The number of ways to choose 3 red balls from the 10 red balls is:

   $\binom{10}{3}$

2. Drawing 3 white balls: The number of ways to choose 3 white balls from the 10 white balls is:

   $\binom{10}{3}$

So, the total number of favorable outcomes (drawing either 3 red or 3 white balls) is:

$\text{Favorable outcomes} = \binom{10}{3} + \binom{10}{3}$

Probability:

The probability of drawing 3 balls of the same color is the ratio of favorable outcomes to the total possible outcomes:

$P(\text{same color}) = \frac{\binom{10}{3} + \binom{10}{3}}{\binom{20}{3}}$

Now, let's calculate the combinations:

$\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$

$\binom{20}{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140$

Thus, the probability is:

$P(\text{same color}) = \frac{120 + 120}{1140} = \frac{240}{1140} = \frac{4}{19}$

Final Answer:

The probability of drawing 3 balls of the same color is $\frac{4}{19}$.

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Would you like any more details or explanations?

Here are a few related questions to explore further:

1. How would the probability change if there were 5 red balls and 15 white balls in the box?
2. What is the probability of drawing exactly 2 red balls and 1 white ball?
3. How would the problem change if we were asked for the probability of drawing 3 balls of different colors?
4. How can the concept of combinations be applied to other probability problems?
5. What is the difference between combinations and permutations in probability?

Tip: When working with probability and combinations, remember that the order in which you draw the balls does not matter, which is why combinations are used instead of permutations.