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A Cow, A Monkey and a Tree





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Q: A Cow is tethered to a tree by a rope that has half the length of the circumference of the tree's stump. A monkey with a stone in hand jumps onto the tree and starts hopping around the tree's branches which covers a circular area of radius five times that of the stump. This causes the cow to run about at random and monkey drops the stone at random. What is the probability that the cow would get hit?

A Perfect Mathematical Christmas Gift

A: The situation can be characterized by the figure shown below (not to scale)


The monkey can be anywhere in the light shaded orange area before it drops the stone (this of course assumes that the stump has no terrace area). The cow, can sweep a spiral as shown in the figure below along either lines (poorly drawn).

To start, lets first compute the area the cow can sweep. Assume the cow moves a bit to the left, the rope would wind up along the stump of the tree. To further formalize this assume it sweeps out an angle \(\theta\) at the center as shown figure below.
In the figure if the cow's rope winds up by a length AB subtending an angle \(\theta\) at the center C, and if the cow moves from D to E an incremental angle of \(d\theta \) is subtended at the centre. The length of the rope is now decreased from \(\pi r\) to \(\pi r - r\theta\). The length of DE can be computed as
$$
(\pi r - r\theta )d\theta
$$
The incremental area of triangle ADE is \(\frac{1}{2}\times DE \times AD\) which works out as
$$
dA = \frac{1}{2}(\pi r - r \theta)^{2}d\theta
$$
To compute the total area, we can integrate out with \(\theta\) running from \([0,\pi]\) and as the area is symmetric, we multiply by 2
$$
A =2 \times \int_{0}^{\pi}\frac{1}{2}r^{2}(\pi - \theta)^{2}d\theta
$$
The above integral simplifies to
$$
A = \frac{r^{2}\pi^{3}}{3}
$$
The total area the monkey can drop the stone is easy to compute as
$$
A_{total} = \pi (5r)^{2} - \pi r^{2} = 24\pi r^{2}
$$
The sought area and probability is simply the ratio of the two areas \(\frac{A}{A_{total}} = \frac{\pi^{3}}{24\times 3} \approx 43\%\)

Some of the best books to own 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.

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

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