A Winning Series

Q: You are required to play a game of skill against opponents A & B. You win if you can win two games in a row. You are presented with two options

1. Play A first, then B followed by A (option A-B-A)
2. Play B first, then A followed by B (option B-A-B)

The probability that you will win against A is \(\frac{1}{3}\) and that against B is \(\frac{1}{2}\). Which option maximizes your chances of winning?

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

A: At first brush it appears that option BAB is your best option because you play B more often and your chances of winning against B are higher than against A. However, there is a surprising and counter intuitive twist to the tale. You win only if you win two games in a row. To understand it better, let us enumerate out the cases for the A-B-A option. The options are shown in the table below

Each row enumerates out a possible scenario. The final column is the probability of that scenario playing out. The underlined ones are the cases when you win the series by winning two games in a row. Likewise, for option B-A-B, the same table works out as shown below

Notice, the probability of winning with the A-B-A option is \(\frac{5}{18}\) vs \(\frac{1}{4}\) for the B-A-B option!

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.