jmc

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 $\left(\genfrac{}{}{0ex}{}{6}{4}\right)=15$ 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 $\left(\genfrac{}{}{0ex}{}{6}{5}\right)=6$ 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 $\frac{24}{243}=\boxed{\frac{8}{81}}$.