### Acute Angles and Wall Clocks

Q: If you were look at a wall clock between 12noon and 1pm at random, what is the probability that the minute hand and the hour hand make an acute angle?
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A: If you rush this one, you will be tempted to answer 50%, but it is not so. Assume we are at 12 noon. The angle swept by the hour hand of a clock in an hour is $$\frac{30^{o}}{60}=(\frac{1}{2})^{o}$$ per minute. If $$t$$ minutes have passed in an hour, the hour hand would sweep out $$\frac{1}{2}t$$ degrees. Likewise the minute hand of the clock sweeps out $$\frac{360^{o}}{60} = 6^{o}$$ per minute. The condition required for an acute angle to form between the hour and minute hand can be phrased as
$$\theta_{m} - \theta_{h} \le 90^{o} \\ 6t - \frac{t}{2} \le 90 \\$$
The above simplifies to
$$t \le \frac{180}{11} \approx 16$$
However, there is one more case that creates an acute angle between the hour and minute hand of the clock. This is when the minute hand makes it ways back to 12. This condition can be stated as
$$270^{o} \le \theta_{m} - \theta_{h} \le 360^{o}\\ 270^{o} \le 6t - \frac{t}{2} \le 360^{o}$$
which in turn implies $$t$$ has a range of $$\frac{2}{11}\times (360 - 270) = \frac{180}{11}$$. The above two inequalities shows that there is a total of $$\frac{2\times 180}{11} = 32.72 \text{ minutes}$$ when the hour hand and the minute hand forms an acute angle. Given that there are 60 minutes in an hour the sought probability is $$\frac{32.72}{60} = 54.53\%$$. Notice, at first blush one is tempted to answer 50% as the sought probability, but when one thinks more about it, it is intuitive to see why it will always be slightly greater than 50%. This is because of the hour hand moves slowly too, maintaining the "acuteness", a bit longer. This gives the higher probability number.

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

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.

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.

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

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 Linear Algebra

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

Linear Algebra (2nd Edition)
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Linear Algebra: A Modern Introduction
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Linear Algebra (Dover Books on Mathematics)
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Linear Algebra Done Right (Undergraduate Texts in Mathematics)
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Linear Algebra, 4th Edition
This is good book …

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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!)
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Econometrics
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