Finding the light bag of coins in one weighing

Question

Ten bags each contain $1{,}000$ coins. In nine bags every coin weighs one ounce; in the remaining bag every coin weighs $10\%$ less. Using a scale that shows exact weight, how can one weighing reveal the light bag?

Short answer

Take $k$ coins from bag $k$ and weigh them together. That is $55$ coins, which would weigh $55$ ounces if all were genuine. If the light bag is bag $k$ , the total falls short by $0.1k$ ounces, so the shortfall times ten names the bag.

Adapted from Crack, Heard on the Street, Q1.55, p.19; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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