Fruit Selection Probability
Pat wants to select 8 pieces of fruit to bring in the car for the people he's driving to Montana with. He randomly chooses each piece of fruit to be an orange, an apple, or a banana. What is the probability that either exactly 3 of the pieces of fruit are oranges or exactly 6 of the pieces of fruit are apples?
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Solution
It is impossible for Pat to select both 3 oranges and 6 apples, so we can calculate the probabilities of these mutually exclusive cases separately and then add to get our final answer. The probability that 3 particular pieces of fruit will be oranges and the rest will not be is given by $ \left(\dfrac{1}{3}\right)^3\left(\dfrac{2}{3}\right)^5=\dfrac{32}{6561} $, and there are $ \binom{8}{3}=56 $ ways of selecting three pieces of fruit to be the oranges so the probability that 3 will be oranges is $ 56\cdot\dfrac{32}{6561}=\dfrac{1792}{6561} $. Similarly, the probability that 6 particular pieces of fruit will be apples and the other two won't be is given by $ \left(\dfrac{1}{3}\right)^6\left(\dfrac{2}{3}\right)^2=\dfrac{4}{6561} $ and there are $ \binom{8}{6}=28 $ ways of selecting which ones will be the apples, so multiplying again gives us a probability of $ 28\cdot\dfrac{4}{6561}=\dfrac{112}{6561} $. Adding those two probabilities give us our final answer: $ \dfrac{1792}{6561}+\dfrac{112}{6561}=\boxed{\dfrac{1904}{6561}} $.
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