Wednesday, November 7, 2012

The Birthday Paradox

The following is one of the most fascinating results from probability and if you are not familiar with it already, it is definitely counter intuitive. So here goes...

Q. There are 10 people in a room. What is the probability that two of them share the same birthday?

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition

A.There are several explanations on the web for this classic puzzle. If there are exactly two people, the probability they have the same birthday is \(\frac{1}{365}\). More importantly the probability that they will have different birthdays is \(1 - \frac{1}{365} = \frac{364}{365}\). For three people in a room, that works out to \(\frac{364}{365}\times \frac{363}{365}\).
Thus the probability that all n people will all have different birthdays is

The probability for n >= 365 is fixed to 1 by the pigeon hole principle. It is interesting to see how the above expression behaves by rearranging some of the terms. The above equation can be rewritten as

Notice the exponential connection to the number of users. It is this general relationship that causes the probability of two users to have the same birthday to rise pretty quickly with the 'n'.

Some books on probability/algorithms
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

Introduction to Probability Theory

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

Let There Be Range!: Crushing SSNL/MSNL No-Limit Hold'em Games
Easily the most expensive book out there. So if the item above piques your interest and you want to go pro, go for it.

Quantum Poker
Well written and easy to read mathematics. For the Poker beginner.

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.

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