Finding the Faulty Coin

 Follow @ProbabilityPuz 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

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 LifeThis 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.
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

The Best Books to Learn Probability

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

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.

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

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.

Fifty Challenging Probl…

The Best Books for Time Series Analysis

If you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.

Pattern Recognition and Machine Learning (Information Science and Statistics)
This is excelle…

The Best Books for Linear Algebra

The following are some good books to own in the area of Linear Algebra.

Linear Algebra (2nd Edition)
This is the gold standard for linear algebra at an undergraduate level. This book has been around for quite sometime a great book to own.

Linear Algebra: A Modern Introduction
Good book if you want to learn more on the subject of linear algebra however typos in the text could be a problem.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Linear Algebra, 4th Edition
This is good book …