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, . Then to 3 significant figures. The Teacher completes the count definition, both parameters and the complement method.
(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 from (i), so , written in the source. Now 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.
Common mistakes
The second binomial count is boxes, and its success probability is the first answer.