A Matter of Confidence

Q: An urn contains 3 red and 3 blue balls. A ball is drawn from and it socked away. Two people now draw balls from it. They record the color and put the ball back into the urn. The first person A, does this 7 times and draws a red ball all 7 times. The second person B, does this 20 times and draws a red ball 14 times. Both conclude the urn has a majority of red balls but who among them have more confidence in their prediction?

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A: One is tempted to assume that A, who has done more draws which strongly indicate it to be a red-ball majority urn is likely to have a higher confidence. But this is not the case when seen in the Bayesian perspective.

Let \(H\) denote the hypothesis that the majority are red balls and \(E\) denote the evidence each user collects. As always, Bayes theorem states
$$
P(H|E) = \frac{P(E|H)P(H)}{P(E|H)P(H) + P(E|\neg H)P(\neg H)}
$$
For user A, the probability that she gets heads 7 times in a row is \(\big(\frac{3}{5})^{7}\) if the urn has a red-ball majority. It is \(\big(\frac{2}{5})^{7}\) if it is a blue-ball majority urn. The prior probability that the urn has a majority of red balls is \(\frac{1}{2}\). Putting this all together gives us the confidence person A has as
$$
P(H|E) = \frac{\big(\frac{3}{5})^{7} \times \frac{1}{2}}{\big(\frac{3}{5})^{7} \times \frac{1}{2}  + \big(\frac{2}{5})^{7} \times \frac{1}{2}}\approx 94.5\%
$$

Likewise for B
$$
P(H|E) = \frac{\big(\frac{3}{5})^{14} \times \big(\frac{2}{5})^{6}\times \frac{1}{2}}{\big(\frac{3}{5})^{14} \times \big(\frac{2}{5})^{6}\times \frac{1}{2}  + \big(\frac{3}{5})^{6} \times \big(\frac{2}{5})^{14}\times \frac{1}{2}}\approx 96.24\%
$$

Notice, B has higher confidence. This is also intuitive when you think more about it. The fact that the balls are being replaced here is what is causing A to have a rarer event happen (i.e. all 7 are drawn red) making it less plausible to draw a firmer conclusion than B.

If you are looking to learn the art of probability here are a few good books to own

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

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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.