Wednesday, January 9, 2013

The Seven Coins of James Bond

Q: James Bond tells you he has seven coins. What is the probability that he has more than a dollar (US dollar)?
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists

A: For starters assume that the coins are equally probable. This simplifies things. The coins could then have values 1c, 5c, 10c and 25c with equal probability. This yields expected value of each coin to be 

$$E = \frac{1}{4}\times ( 1 + 5 + 10 + 25) \\ = 10.25c $$

Therefore, the expected value of seven such coins would be \( 7 \times 10.25c = 71.75c \). To proceed we can make a simplifying assumption that the distribution of expected value of a coin follows a normal distribution. Note, this is not entirely accurate, but it is something we can work with.

First compute the variance of the values. The variance of a random variable is the square of the standard deviation given as \(\frac{\sum_i (x_i - \mu)^{2}}{4}\). If you plug in the values you get

$$ \frac{1}{4}\times{ (1 - 10.25)^{2} + (5 - 10.25)^{2} + (10 - 10.25)^{2} + (25 - 10.25)^{2} }  = 82.68$$

As we are adding up seven such numbers, the variances are also additive. This yields a total variance of \(7 \times 82.68 =578.76\).
We now have a nice Gaussian distribution with expected mean 71.75 and variance 578.76.

For any normal distribution, given its mean and variance, we can compute the area under the probability density function curve which is greater than a given amount. For example the figure below shows a simulation of the problem in R.
The area in red divided by the total area is the sought probability.

To compute the probability we need to compute the z score for the sought value (100). This is simply the number of (floating point) standard deviations from the mean given as
$$z = \frac{x - \mu}{\sqrt{\sigma^{2}}}$$
The z-score for this particular problem works out to \(z = 1.17\). The required area can be got from any of the standard tables or in any R package, it works out to 12% which is the sought probability.

An alternate solution is to comprehensively count the number of cases when 7 coins add up to greater than 100. In this approach there are a total of \(4^7\) possible cases. A quick run of a recursive Perl function returns the count as 2262. This yields a true probability estimate of \(\frac{2262}{4^7} = 13.8\%\). So our estimate is close, but not entirely accurate. The Perl function / script used is shown below.

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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