The "Unconscious Statistician"

Q: You scoop up a small, uniformly random, volume \(V\) of sand from a container containing \(V_0\) amount in volume of sand and drop it on a square plate of side \(a\) such that it makes a cone of height \(h\). Next, you drop a small ball vertically somewhere on the plate. What is the average probability that it falls on the cone of sand? (see fig below)

A Book on Statistical Inference

A: First off, the title of this post is not an eye catching phrase. There does exist a "law" in statistics that goes by the name "the law of the unconscious statistician". Googling for that term will yield results! It has to do with the following scenario. Assume you have a random variable \(X\) which follows some distribution \(f(X)\). You want to find the expected value of some other function which takes \(X\) as an argument, call this function \(g\). The law of the unconscious statistician says that you need not find \(g(X)\) explicitly. Instead, you can get to the expected value by simply evaluating
$$
E(g(X)) = \int_{-\infty}^{+\infty}g(x)f(x)dx
$$
Now, back to the problem. The cone of sand, when seen from above is just a circular area. The volume of the cone of height \(h\) and base radius \(r\) is given by
$$
V = \frac{1}{3}\pi r^{2}h
$$
rearranging for the area \(\pi r^{2}\) yields
$$
\pi r^{2} = \frac{3V}{h}
$$
The probability (as a function of \(V\)) that a small ball dropped vertically would land on that cone of sand is the ratio of the area of the circle \(\pi r^{2}\) to that of the square plate \(a^{2}\) which is given by
$$
P = \frac{\pi r^{2}}{a^{2}} = \frac{3V}{a^{2}h}
$$
This is a simple linear function of \(V\). Also note that \(V\) is a uniformly distributed between \([0,V_{0}]\), meaning it can take a value in that range with equal probability. This yields a probability density function for \(V\) as \(\frac{1}{V_{0}}\). The probability that we are after is a function of \(V\). To compute average value of this probability, we use the law of the unconscious statistician as follows
$$
E[P(V)] = \int_{0}^{V_{0}} \frac{3V}{a^{2}h}\times\frac{1}{V_{0}}dV
$$
which works out to
$$
E[P(V)] = \frac{3V_{0}}{2a^{2}h}
$$

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.