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?
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 $\frac{24}{243}=\boxed{\frac{8}{81}}$.