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### Polya's Urn

Q: An urn has $$r$$ red balls and $$b$$ blue balls. Someone draws a ball at random, its colour observed and put back into the urn. You do not know what was observed. However that person puts back $$x$$ balls of the same colour back into the urn. Now, you draw a second ball from this urn. What is the probability that it is red?

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

A: The framing of this puzzle follows directly from the "Polya's Urn" process. It presents yet another surprising result from Bayesian reasoning. Intuitively, it appears that the act of putting in new balls of the same colour would tamper with the probability of drawing a red ball for the second draw. But does it? Lets take a look.

The probability that a red ball is drawn from the urn in the first draw is $$\frac{r}{r+b}$$ and for a blue ball would be $$\frac{b}{r+b}$$. The second draw, if it is a red ball, could be a consequence of either a red ball being drawn the first time or a blue ball.

For the second draw, the probability that a red ball is drawn if a red ball is drawn the first time, would be $$\frac{r+x}{r + b + x}$$. The probability that a red ball is drawn if a blue ball is drawn the first time, would be $$\frac{b}{r+b+x}$$. This layout is shown in the figure below.

The probability that a red ball is drawn on the second draw is
$$P(\text{Red: Draw=2})=\frac{r + x}{r + b + x}\times\frac{r}{r+b} + \frac{r}{r+b+x}\times\frac{b}{r+b}$$
The above simplifies as
$$\frac{(r+b+x)r}{(r+b+x)(r+b)} = \frac{r}{r+b}$$
Note, the probability remains exactly the same!
As with most Bayesian stuff, the result eventually becomes intuitive when we spend more time thinking of the problem. The act of adding $$x$$ balls based on the outcome of the first draw, is really meaningless!

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.

### 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 Three Magical Boxes

Q: You are playing a game wherein you are presented 3 magical boxes. Each box has a set probability of delivering a gold coin when you open it. On a single attempt, you can take the gold coin and close the box. In the next attempt you are free to either open the same box again or pick another box. You have a 100 attempts to open the boxes. You do not know what the win probability is for each of the boxes. What would be a strategy to maximize your returns?

Machine Learning: A Probabilistic Perspective (Adaptive Computation and Machine Learning series)

A: Problems of this type fall into a category of algorithms called "multi armed bandits". The name has its origin in casino slot machines wherein a bandit is trying to maximize his returns by pulling different arms of a slot machine by using several "arms". The dilemma he faces is similar to the game described above. Notice, the problem is a bit different from a typical estimation exercise. You co…

### The Best Books for Time Series Analysis

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!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.

Pattern Recognition and Machine Learning (Information Science and Statistics)
This is excelle…