Four green balls and five orange balls are placed in a bag. What is the probability that you randomly draw three of the same color? Enter your answer as a fraction a/b in simplest form.

We have a total of 4 green balls and 5 orange balls, making a total of \( 4 + 5 = 9 \) balls. We need to calculate the probability of randomly drawing three balls of the same color.

### Two possible cases:
1. All three balls are green.
2. All three balls are orange.

### Case 1: Drawing three green balls
To calculate the number of ways to draw 3 green balls out of 4, we use the combination formula:
\[
\binom{4}{3} = \frac{4!}{3!(4-3)!} = 4
\]

### Case 2: Drawing three orange balls
Similarly, to calculate the number of ways to draw 3 orange balls out of 5:
\[
\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
\]

### Total number of favorable outcomes:
The total number of favorable outcomes (either 3 green or 3 orange) is:
\[
4 + 10 = 14
\]

### Total number of possible outcomes:
The total number of ways to choose any 3 balls from the 9 is:
\[
\binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84
\]

### Probability:
The probability of drawing three balls of the same color is:
\[
\frac{14}{84} = \frac{1}{6}
\]

Thus, the probability is \( \frac{1}{6} \).

Would you like more details on any of the steps? Here are 5 related questions for further practice:

1. What is the probability of drawing three orange balls?
2. What is the probability of drawing two green balls and one orange ball?
3. How would the probability change if there were 6 orange balls instead of 5?
4. If you draw two balls, what is the probability that both are the same color?
5. How many different ways are there to draw three balls in total?

**Tip:** When solving probability problems involving combinations, make sure to clearly identify the total number of possible outcomes first.

Discover the probability of drawing three balls of the same color from a bag containing four green balls and five orange balls. This detailed solution explains how to apply combinatorics and probability formulas to solve the problem. Suitable for students in Grades 9-12.

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