Exactly one bruised apple in four

Question

A crate holds 10 apples, of which 3 are bruised and 7 are fine. You grab 4 apples at random in one scoop. What is the probability that exactly 1 of your 4 apples is bruised?

Accepted Answer

Order does not matter for a single scoop, so count groups with combinations, where \binom{n}{k} is the number of ways to choose k from n. Total ways to choose 4 apples from 10: \binom{10}{4}=210. Favorable: exactly 1 bruised and 3 fine. Pick 1 of the 3 bruised and 3 of the 7 fine: \binom{3}{1}	imes\binom{7}{3}=3	imes 35=105. Probability: \frac{105}{210}=\frac{1}{2}. A clean fifty-fifty. Takeaway. Choose the wanted type from its pile and the remainder from the rest, multiply, then divide by every way to grab the handful. Here the counts happen to land on exactly one-half.

Exactly one bruised apple in four

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