Sarah buys $3$ gumballs from a gumball machine that contains $10$ orange, $6$ green, and $9$ yellow gumballs. What is the probability that the first gumball is orange, the second is green or yellow, and the third is also orange?
1. **Calculate the probability that the first gumball is orange:**
   - There are 10 orange gumballs out of a total of 25 gumballs.
   - The probability of drawing an orange gumball first is:
     \[
     P(\text{First is orange}) = \frac{10}{25} = \frac{2}{5}
     \]

2. **Calculate the probability that the second gumball is green or yellow:**
   - After drawing the first orange gumball, there are now 24 gumballs left.
   - The number of green gumballs is 6, and the number of yellow gumballs is 9.
   - The probability of drawing a green gumball second is:
     \[
     P(\text{Second is green}) = \frac{6}{24} = \frac{1}{4}
     \]
   - The probability of drawing a yellow gumball second is:
     \[
     P(\text{Second is yellow}) = \frac{9}{24} = \frac{3}{8}
     \]
   - The probability that the second gumball is either green or yellow is:
     \[
     P(\text{Second is green or yellow}) = P(\text{Second is green}) + P(\text{Second is yellow}) = \frac{1}{4} + \frac{3}{8} = \frac{2}{8} + \frac{3}{8} = \frac{5}{8}
     \]

3. **Calculate the probability that the third gumball is orange:**
   - After drawing the first orange gumball and a second gumball that is either green or yellow, there are now 23 gumballs left.
   - The number of remaining orange gumballs is 9.
   - The probability of drawing an orange gumball third is:
     \[
     P(\text{Third is orange}) = \frac{9}{23}
     \]

4. **Combine the probabilities:**
   - The combined probability of drawing an orange gumball first, a green or yellow gumball second, and an orange gumball third is:
     \[
     P(\text{First is orange}) \times P(\text{Second is green or yellow}) \times P(\text{Third is orange}) = \frac{2}{5} \times \frac{5}{8} \times \frac{9}{23}
     \]
   - Simplify the expression:
     \[
     \frac{2}{5} \times \frac{5}{8} \times \frac{9}{23} = \frac{2 \times 5 \times 9}{5 \times 8 \times 23} = \frac{90}{920} = \frac{9}{92}
     \]

The final answer is \(\boxed{\frac{9}{92}}\).