+150 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?
Thank you!
+6248 \(\text{the two events are distinct so the probability of their union is the sum of their individual probabilities}\)
\(\text{First we compute how many total fruit arrangements there are}\\ \text{This is equivalent to the stars and bars problem of sorting 8 fruits into 3 bins}\\ \text{Thus there are }\\ n=\dbinom{8+3-1}{3-1} = \dbinom{10}{2} = 45\)
\(\text{Next we compute how many arrangements have exactly 3 oranges}\\ \text{Choosing 3 oranges we have 5 fruits left to choose among two choices}\\ \text{This is again the stars and bars problem now putting 5 fruits into 2 bins}\\ n_{3o} = \dbinom{5+2-1}{2-1} = \dbinom{6}{1}=6\\ P[3o] = \dfrac{6}{45} = \dfrac{2}{15}\)
\(\text{more quickly this time}\\ P[6a] = \dfrac{\dbinom{2+2-1}{2-1}}{45} = \dfrac{1}{15}\)
\(P[3o \cup 6a] = P[3o]+P[6a] = \dfrac{2}{15}+\dfrac{1}{15} = \dfrac{1}{5}\)
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