Question:
Chris wants to put six plants in a row on his windowsill. He randomly chooses each plant to be an aloe plant, a basil plant, or a violet. What is the probability that either exactly four of the plants are aloe plants or exactly five of the plants are basil plants?

Answer:
It is impossible for Chris to have both four aloe plants and five basil plants, so first we consider the case of four aloe plants. There are $\binom{6}{4}=15$ ways to choose which of the plants are the aloe plants. For each of these choices, there is a $\left( \frac{1}{3} \right)^4 \left( \frac{2}{3} \right) ^2$ chance of that choice happening. Therefore, the total probability that Chris chooses exactly four aloe plants is $15\cdot\left( \frac{1}{3} \right)^4 \left( \frac{2}{3} \right) ^2=\frac{20}{243}$. There are $\binom{6}{5}=6$ ways to choose five plants to be the basil plants. For each of these choices, there is a $\left( \frac{1}{3} \right)^5 \left( \frac{2}{3} \right) ^1$ chance of that choice happening. Therefore, the total probability that Chris chooses exactly five basil plants is $6\left( \frac{1}{3} \right)^5 \left( \frac{2}{3} \right) ^1=\frac{4}{243}$. The probability that Chris chooses either four aloe plants or five basil plants is $\frac{24}{243}=\boxed{\frac{8}{81}}$.