A Bayesian Treasure Hunt

Follow @ProbabilityPuz

Q: Four friends A,B,C,D learn that a square tract of land they own has a treasure buried somewhere. They divide the tract of land into four equal quadrants, estimate that the probability in each of those square tracts as \(\{r,r,r,p\}\). A starts digging his quadrant first. The probability that A might miss the treasure given the dig is \(q\). Having dug out his quadrant, A exclaims that he still hasn't found the treasure. What is the updated probabilities of

1. A's quadrant has the treasure?
2. B's quadrant has the treasure?

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)

A: This is a classic Bayesian puzzle. The fact that A has dug up his quadrant and revealed that there is no treasure leaves open the possibility that he might have missed the treasure. Yet it is information that must be worth something which will help B,C,D update their belief that the treasure is present in their quadrant. First off lets calculate A's updated belief. A's prior is \(p\). Let \(T\) denote the event that the treasure is in A's quadrant given nothing showed up during the dig. Let \(M\) denote the event that A's dig resulted in a miss. We know from Bayes theorem
$$
P(T|M) = \frac{P(M|T)P(T)}{P(M)}
$$
\(P(M)\) can occur in two ways. Either, the treasure just isn't there or it was there and A missed it. So, \(P(M)\) works out as
$$
P(M) = (1-q)\times p + (1 - p) = 1 - pq
$$
\(P(M|T) = 1 - q\) and \(P(T)=p\). So A's updated belief that the treasure still resides in his quadrant is
$$
P(T|M) = \frac{p(1-q)}{1 - pq}
$$
As far as B is concerned, let \(T_{B}\) denote the event that the treasure is in his quadrant. Casting this in the Bayesian framework yields
$$
P(T_{B}|M) = \frac{P(M|T_{B}) P(T_{B})}{P(M)}
$$
Note, the denominator remains the same. The numerator works out as: \(P(M|T_{B}) = 1\) and \(P(M) = r\). So the update rule for B (and for that matter any other person) is
$$
P(T_{B}|M) = \frac{r}{1 - pq}
$$

Check out 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
Overall an excellent book to learn probability, well recommended for undergrads and graduate students

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.

Introduction to Probability, 2nd Edition
A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.

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 and even 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
This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"

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.

A Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

Probability and Statistics (4th Edition)This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good