Acute Angles and Wall Clocks

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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)
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Introduction to Algorithms
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Introduction to Probability Theory
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An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
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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)
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Understanding Probability: Chance Rules in Everyday Life
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Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)
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Discovering Statistics Using R
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A Course in Probability Theory, Third Edition
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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