Swapping Colored Balls from Bags

Q: Alice has a bag with two red balls and a blue ball. There is another bag with two blue balls and one red ball. A ball is chosen randomly from Alice's bag and placed in the other bag. Next a random ball is chosen from that other bag and put back in Alice's bag. What is the probability that the remainder of the balls in her bag are red if

1. Alice draws a ball from her bag notices it to be a red ball
2. Alice draws a ball from her bag and notices it to be a blue ball

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
A: If Alice draws a red ball, the only way the other two balls will be red balls is if in the first draw a blue ball was chosen and in the next draw a red ball was chosen. This happens with probability \(\frac{1}{3}\) and \(\frac{1}{2}\) respectively. This yields an overall probability of \(\frac{1}{3}\times\frac{1}{2}=\frac{1}{6}\)

If the ball Alice draws is blue, there are only two scenarios that could have caused this.

1. A red ball was drawn at the first draw from the bag and a red ball returned.
2. A blue ball was drawn at the first draw from the bag and a blue ball returned.

Note that any other scenario would not result in two red balls to remain in the bag while simultaneously being able to draw a blue ball. The probability of scenario 1) is \(\frac{2}{3}\times\frac{1}{2}\times\frac{1}{3}=\frac{1}{9}\) and the probability of scenario 2) is \(\frac{1}{3}\times\frac{3}{4}\times\frac{1}{3}=\frac{1}{12}\). As the scenarios are independent, the sought probability is \(\frac{1}{9} + \frac{1}{12} = \frac{7}{36}\)

Some good books on the art of probability
Introduction to Probability Theory

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.