Finding the Faulty Coin

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Q: You have a large number of coins out of which some small fraction \(p\) are faulty ones. The faulty ones have slightly different weight (maybe more or less) and for some reason the error-free-test to weigh the coins is expensive and you want to minimize the number of tests done. So you embark on weighing them in bulk lots. What is the optimal lot size so that you minimize the number of weighs per coin.

Probability Theory: The Logic of Science
A: Lets assume we went with a lot size of \(x\). If we can compute the expected number of weighs per coin needed for a lot size of \(x\) then this strategy must apply to the entire set of coins. So we need not know the entire lot size.

Next, let us try and estimate the expected number of weighs needed. There are two scenarios.

1. You weigh the entire lot and you find no discrepancy in the weight
2. You weigh the entire lot and you find a discrepancy in the weight.

Next we note the following

1. The probability that any one coin is faulty is \(p\)
2. The probability that any one coin is not faulty is \(1 - p\). 
3. The probability that all coins are not faulty is \((1-p)^{x}\).
4. The probability that at least one coin is faulty is \(1 -  (1-p)^{x}\)

The expected number of tests is at least 1 per lot. However, with probability \(1 -  (1-p)^{x}\) you would have to weight all the \(x\) coins. This results in a total expectation of
$$
E(\text{tests}) = 1 + x\big[1 -  (1-p)^{x}]
$$
The expected number of tests per coin is given by
$$
E(\text{tests/coin}) = \frac{1}{x} + 1 -  (1-p)^{x}
$$
If we know that \(p\) is small, you can approximate the second term in the above equation as
$$
(1-p)^{x} \approx 1 - xp
$$
Which simplifies the expected number of weighs per coin as follows
$$
E(\text{tests/coin}) = \frac{1}{x} + xp \\
$$
Differentiating the above w.r.t. \(x\) and setting it to 0 yields
$$
\frac{dE}{dx} = -\frac{1}{x^2} + p = 0
$$
Implying
$$
x_{opt} = \frac{1}{\sqrt{p}}
$$

If you are looking to learn probability here are some good books to own

Book	Notes/Comments
Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)	This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.
Introduction to Algorithms	This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists
Introduction to Probability Theory	Overall an excellent book to learn probability, well recommended for undergrads and graduate students
An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition	This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)	A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.
Introduction to Probability, 2nd Edition	A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.
Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)	An excellent resource (students, engineers and even entrepreneurs) if you are looking for some code that you can take and implement directly on the job
Understanding Probability: Chance Rules in Everyday Life	This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"
Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)	This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.
Discovering Statistics Using R	This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.
A Course in Probability Theory, Third Edition	Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online
Probability and Statistics (4th Edition)	This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good