### 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 Cha

### Fun with Uniform Random Numbers

Q: You have two uniformly random numbers x and y (meaning they can take any value between 0 and 1 with equal probability). What distribution does the sum of these two random numbers follow? What is the probability that their product is less than 0.5. The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists A: Let z = x + y be the random variable whose distribution we want. Clearly z runs from 0 to 2. Let 'f' denote the uniform random distribution between [0,1]. An important point to understand is that f has a fixed value of 1 when x runs from 0 to 1 and its 0 otherwise. So the probability density for z, call it P(z) at any point is the product of f(y) and f(z-y), where y runs from 0 to 1. However in that range f(y) is equal to 1. So the above equation becomes From here on, it gets a bit tricky. Notice that the integral is a function of z. Let us take a look at how else we can simply the above integral. It is easy to see that f(z-y) = 1 when (

### The Best Books for Linear Algebra

The following are some good books to own in the area of Linear Algebra. Linear Algebra (2nd Edition) This is the gold standard for linear algebra at an undergraduate level. This book has been around for quite sometime a great book to own. Linear Algebra: A Modern Introduction Good book if you want to learn more on the subject of linear algebra however typos in the text could be a problem. Linear Algebra (Dover Books on Mathematics) An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book. Linear Algebra Done Right (Undergraduate Texts in Mathematics) A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra. Linear Algebra, 4th Ed