Tuesday, December 25, 2012

The Magician's Coin Tossing Experiment

Q: A magician calls you in and using his magical powers claims to make all coin tosses he does fall a heads. You challenge him, and he tosses the coin once and the coin falls heads. What is the probability that he actually has magical powers? He tosses it again, and it falls heads. What is the probability now that he actually has magical powers?

A: This is classic Bayesian. Assume you have a low belief in this person. Set it as 'p'. He flips the first coin and it lands a heads. This new evidence, which favours the hypothesis must lead us to adjust our belief in our value of 'p'. This may not seem intuitive at first glance, partly because the amount of information is very small (1 toss). But lots of real world problems deal with scarce data (more on that on a later post).

Here is an earlier post describing Bayesian reasoning. So, to cast it in that same framework lets start with our hypothesis.

Hypothesis H = Magical Powers Exist
Evidence    E  = 1 heads seen on a coin toss
Prior               = p

So this gets cast as

We also know the following,
  1. If H is true, then P(E|H) must equal 1. 
  2. P(H) equals 'p'.
  3. If H is not true, then P(E|-H) = 1/2
  4. P(-H) = 1 -p
The above equation now translates to

which further simplifies to

You can plug in anything for 'p', your prior belief. If you believe it to be 1/2, then your new probability is 2/3.

So what happens at the next toss which comes up heads. The prior for this case is 2/3. Plug this in, and it yields 4/5. You can keep updating this. If you chart it out in R, you can see that this number will eventually converge to 1, but will never be 1. That is the beauty of the Bayesian estimate. See figure below for how this belief varies with each successive toss of heads starting with a prior of 0.01.

The R code that created the data above is fairly simple.

If you are looking to buy some books in probability here are some of the best 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

Introduction to Probability Theory

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

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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