Problem 6 asks for the smallest number of draws. Here it is worked on the guide's sibling: 500 people, 50 diseased, target 0.95.
Let them predict first.
Reveal: 'none' is one chain of healthy draws, so use the complement (Lemma 1): P(at least one) = 1 - P(none).
Draw one person at a time. Before draw j + 1, j healthy people are gone.
Ordered draws or unordered subsets give the same probability.
Let them predict first.
Reveal: m = 27 gives 0.9464 < 0.95 and m = 28 gives 0.9520 >= 0.95, so the smallest m is 28. Always check m - 1.
Let them predict first.
Reveal: 0.9^m <= 0.05, take logs, divide by ln 0.9 < 0 and flip: m >= 28.43, so m = 29.
Without replacement each healthy draw leaves a slightly higher diseased share.
Five moves. Then do the guide's drills by hand.