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Worked solution

Count qualifying chocolates, then qualifying boxes

A factory packs chocolates into boxes of 20. A chocolate fails the minimum packing criterion if it weighs less than 20 grams. On average 2% of chocolates fail. (i) Find the probability that a randomly chosen box contains at least one failing chocolate. (ii) For four randomly chosen boxes, find the probability that exactly two contain at least one failing chocolate.

CJC · Lecture · Example 3

(i) Count failing chocolates within one box

Let X count chocolates, out of 20, that fail the minimum criterion. Under the source constant-probability independent-chocolate model, X∼B(20,0.02)X\sim B(20,0.02). Then P(X≥1)=1−P(X=0)=1−(0.98)20=0.33239…=0.332P(X\ge1)=1-P(X=0)=1-(0.98)^{20}=0.33239\ldots=0.332 to 3 significant figures. The Teacher completes the count definition, both parameters and the complement method.

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(ii) Count boxes satisfying the first event

Let Y count boxes, out of 4, containing at least one failing chocolate. Use the independent-box model with success probability q=1−(0.98)20q=1-(0.98)^{20} from (i), so Y∼B(4,q)Y\sim B(4,q), written B(4,0.33239)B(4,0.33239) in the source. Now P(Y=2)=(42)q2(1−q)2=0.29545…=0.295P(Y=2)=\binom42q^2(1-q)^2=0.29545\ldots=0.295 to 3 significant figures. This retains the Teacher count/parameter/probability completions. The source precision remark says to keep at least 5 significant figures in intermediate probabilities before a final 3-significant-figure answer; using the stored unrounded q is preferable.

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Common mistakes

The second binomial count is boxes, and its success probability is the first answer.

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