Santa's Dice Game

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Q: Santa offers you to play a game of dice. You get to roll a dice six times. You can stop rolling whenever you wish and you get the dollar amount shown on that roll. What is an optimal strategy to maximize your payoff?

Probability and Statistics (4th Edition)


A: Let us take a moment and think through this. At each point in the sequence of rolls you make, you have a decision to make. Do you keep rolling or do you stop and walk away with what is being "offered" to you? You also need to bear in mind that if you keep pushing your luck you will reach a point (the 6th roll) where you would have to be content with whatever comes out for the last roll. So lets start with the simple case of what the expected payoff is for the last roll. Lets call this \(E_{6}\). To compute it, simply take the payoff multiplied by the respective probability.
$$
E_{6} = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{7}{2} = 3.5
$$
The general strategy to be followed is to check what the expected pay off is for the next roll and accept nothing less than that. Working backwards, lets consider each of the rolls starting with the 5th roll.

Roll 5:
You know that on 6th roll you get an expected payoff of 3.5. You should accept anything greater than 3.5, which means you stop rolling if you get a 4,5 or 6, put in other words anything greater than 4. The expected payoff is
$$
\frac{4 + 5 + 6}{6} + \frac{7}{2}\times\frac{1}{2} = \frac{17}{4}
$$
Roll 4:
Looking ahead you find the expected payoff for the 5th roll is \(\frac{17}{4} = 4.25\). You stop if you get a 5 or greater for this roll. The expected payoff would be
$$
\frac{5+6}{6} + \frac{17}{4}\times\frac{4}{6} = \frac{14}{3}
$$

Roll 3:
The expected payoff from roll 4 above is \(\frac{14}{3} = 4.66\) which means you stop if you get a 5 or above. The expected payoff for this roll is
$$
\frac{5 + 6}{6} + \frac{14}{3}\times\frac{4}{6} = \frac{89}{18}
$$

Roll 2:
The expected payoff from roll 3 above is \(\frac{89}{18}=4.94\) implying accept and stop if roll gives 5 or 6. The expected payoff for this roll is
$$
\frac{5 + 6}{6} + \frac{89}{18}\times\frac{4}{6} = 5.12
$$

Roll 1:
Being the first roll, your expected payoff from any subsequent rolls is 5.12. So you should accept nothing short of a 6 from the first roll.

The optimal strategy would then be to stop at each of the first five rolls (when you have the option to stop) if you get 6,5,5,5,4 respectively. An interesting takeaway here is the expected gains from following this strategy can be computed as
$$
\frac{1}{6}\times 6 + \frac{5}{6}\times 5.12 = 5.26
$$
which is quite high for a seemingly random game!

If you are looking to buy some books in probability here are some of the best 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

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