### Maximizing Chances in an Unfair Game

Q: You are about to play a game wherein you flip a biased coin. The coin falls heads with probability $$p$$ and tails with $$1 - p$$ where $$p \le \frac{1}{2}$$. You are forced to play by selecting heads so the game is biased against you. For every toss you make, your opponent gets to toss too. The winner of this game is the one who wins the toss the most. You, however get to choose the number of rounds that get played. Can you ever hope to win?

Machine Learning: The Art and Science of Algorithms that Make Sense of Data

A: At a first look, it might appear that the odds are stacked against you as you are forced to play by choosing heads. You would think that your chances or winning decrease as you play more and more. But, surprisingly there is a way to choose the optimal number of tosses (remember, you get to choose the number of times this game is played). To see how, lets crank out some numbers. If you get to toss the coin $$n$$ times, then the total number of coin tosses you and your opponent flips is $$2n$$.  Out of the $$2n$$ tosses if $$y$$ turns out heads, the probability that you would win is
$$P(\text{y Wins}) = {2n \choose y} p^{y}(1 - p)^{2n - y}$$
In order to win, the value of $$y$$ should run from $$n + 1$$ to $$2n$$ and the overall probability works out to
$$P(\text{Win}) = \sum_{y = n + 1}^{2n}{2n \choose y} p^{y}(1 - p)^{2n - y}$$
We can work out the probability of winning by choosing various values of $$p$$ and $$n$$ and chart them out. Here is the R code that does it.

The code runs pretty quickly and uses the data.table package. All the processed data is contained in variables z and z1. They are plotted using the ggplot package to generate the following charts for the strategy.

The first chart shows the variation of the probability of winning by the number of games played for various probability bias values.

The next chart shows the optimal number of games to play for a given bias probability value.

Some good books to own for learning probability is listed here
Yet another fascinating area of probability are Monte Carlo methods. Here are a list of good books to own to learn Monte Carlo methods.

### 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

### 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

### 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 (