Plant Selection Odds
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?
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Solution
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}} $.
Freeflows