Q: There are four integer numbers chosen purely randomly and added up. You are told that the resulting number is even. What is the probability that all the original numbers were even?

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

A: If we did not know any information about the resulting number, then the probability of all numbers being even is simply \(\frac{1}{16}\) as there are \(2^{4} = 16\) ways for choosing 4 numbers as either odd or even. However we do know that their sum is even.

In addition to the above the following also holds.

Since the order of addition does not matter, the above results yields the table below

If we assume "4E" to be the hypothesis that all four numbers are even and "E" be the event that the final sum is even, we can cast the problem in a Bayesian framework as follows

$$ P(4E|E) = \frac{P(E|4E)\times P(4E)}{P(E|4E)\times P(4E) + P(E|2E)\times P(2E) + P(E|0E)\times P(0E)}$$

Notice we have eliminated cases where \(P(E|3E)\) & \(P(E|1E)\) get set to 0. So the above works out to

$$ P(4E|E) = \frac{1\times \frac{1}{16}}{1\times \frac{1}{16} + 1\times \frac{6}{16} + 1\times \frac{1}{16}} = \frac{1}{8}$$

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

A: If we did not know any information about the resulting number, then the probability of all numbers being even is simply \(\frac{1}{16}\) as there are \(2^{4} = 16\) ways for choosing 4 numbers as either odd or even. However we do know that their sum is even.

In addition to the above the following also holds.

- The sum of two even numbers is always even
- If \(x_1 = 2n + 2\) is an even number and \(x_2 = 2m + 2\) is another even number, their sum is \(2(n+m) + 4\) which is even.
- The sum of two odd numbers is also always even
- If \(x_1 = 2n + 1\) is an odd number and \(x_2 = 2m + 1\) is another odd number, their sum is \(2(n+m) + 2\) which is even.
- The sum of an odd and an even number is always odd
- If \(x_1 = 2n + 1\) is an odd number and \(x_2 = 2m + 2\) is an even number their sum is \(2(n+m) + 3\) which is always odd.

Since the order of addition does not matter, the above results yields the table below

$$ P(4E|E) = \frac{P(E|4E)\times P(4E)}{P(E|4E)\times P(4E) + P(E|2E)\times P(2E) + P(E|0E)\times P(0E)}$$

Notice we have eliminated cases where \(P(E|3E)\) & \(P(E|1E)\) get set to 0. So the above works out to

$$ P(4E|E) = \frac{1\times \frac{1}{16}}{1\times \frac{1}{16} + 1\times \frac{6}{16} + 1\times \frac{1}{16}} = \frac{1}{8}$$

Some must buy books on 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

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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