Escaping from a Forest

Q: You are stuck in a forest. You have no information whatsoever on where you are in the forest, however you do know that the forest is shaped in the form of a very long rectangular strip of width \(b\). You decide to walk out of the forest. What strategy would you adopt? If you were on the edge, what is the average expected distance you would walk? Assume you can measure what distance you can walk and can hold the orientation.

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists

A: A reasonable escape strategy is a fairly simple one. Since you are surrounded by the forest you do not know your orientation w.r.t the forest (see fig). Put in other words, you do not know \(\theta\).

You pick a direction to walk and walk up a distance \(b\). Once you have walked that distance you make a \(90^{o}\) turn to the right and continue to walk a distance (of at most) \(b\). Using this strategy it is guaranteed that you would hit the boundary of the forest strip!
To Compute the average distance travelled assuming you are already at the edge of the forest strip you can proceed as follows: from the figure above, we can see that \(\angle BAC = \angle DBE\) and
$$
\overline{AB} = b \\
\overline{BC} = b \sin(\theta) \\
\overline{DB} = b - b\sin(\theta)\\
\overline{BE} = \frac{b - b\sin(\theta) }{\cos(\theta)}
$$
The distance travelled assuming you are at the edge of the forest is
$$
d = \overline{AB} + \overline{BE}
$$

Assuming \(b=1\) this simplifies to
$$
d = 1 + \frac{1 - \sin(\theta)}{\cos(\theta)}
$$
The above function only covers the case when \(\theta\) runs between \([0,\frac{\pi}{2}]\). For the case when \(\theta\) runs from \([\frac{\pi}{2},\pi]\) the corresponding function works out to
$$
d = 1 - \tan\theta
$$

Trigonometry

The average distance you would expect to walk is the average of the above function \(d\) with \(\theta\) running from \([0,\pi]\). This in turn is given by
$$
d = \frac{2}{\pi}\big( \int_{0}^{\pi} 1 + \frac{1 - \sin \theta}{\cos \theta}d\theta + \int_{\frac{\pi}{2}}^{\pi} 1 - \tan\theta d\theta \big)
$$

Here is a script in R that evaluates this integral.
Discovering Statistics Using R

#!/usr/bin/Rscript

mval = 0
count = 0
for(theta in seq(0,pi/2-0.001,by=0.001)){
  t1 = 1 + ((1 - sin(theta))/cos(theta))
  mval = mval + t1
  count = count + 1
}
for(theta in seq(pi/2+0.001,pi,by=0.001)){
  t1 = 1 - tan(theta)
  mval = mval + t1
  count = count + 1
}
mval = mval / count
cat("Mean Value = ",mval,"\n")

The above script yields a value of 3.6. Also note, the above summation runs from \([0,\pi]\) whereas in reality it should run from \([0,2\pi]\). But the range from \([\pi,2\pi]\) the distance function is practically 0. So the average value would work out to \(\frac{3.6 + 0}{2} = 1.8\)

Some good 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

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)

Introduction to Probability, 2nd Edition

The Mathematics of Poker
Good read. Overall Poker/Blackjack type card games are a good way to get introduced to probability theory

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/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 A bit pricy when compared to the first one, but I like the look and feel of the text used. It is simple to read and understand which is vital especially if you are trying to get into the subject

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.