Q: You are given two urns and 5 black and 5 white balls. An urn will be picked at random and a ball drawn from it. If the ball is white you win. How would you distribute the balls so as to maximize the chances of winning?
The Moscow Puzzles: 359 Mathematical Recreations (Dover Recreational Math)
A: Put in 1 white ball in one urn and the remaining 4 and the other 5 black balls in the other urn. This maximizes your chances of winning. If the urn with the 1 white ball is chosen you win with 100% probability, if the other urn is chosen you win with probability = \(\frac{4}{9}\). This adds up to
$$ P(win) = \frac{1}{2}\times 1 + \frac{1}{2}\times\frac{4}{9} = \frac{13}{18} $$
You can extend this recursively just for kicks. For example if every urn had two urns inside them and you had to fill them up with the balls to maximize you chances how would you do it? It is the same logic, one white ball in each and the last one should have all the remaining (shown in figure)
In this case your win chances come up to 0.82
$$P(win) = \frac{1}{4}\times 1 + \frac{1}{4}\times 1 + \frac{1}{4}\times 1 + \frac{1}{4}\times\frac{2}{7} = 0.82$$
Here are 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
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
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.
The Moscow Puzzles: 359 Mathematical Recreations (Dover Recreational Math)
A: Put in 1 white ball in one urn and the remaining 4 and the other 5 black balls in the other urn. This maximizes your chances of winning. If the urn with the 1 white ball is chosen you win with 100% probability, if the other urn is chosen you win with probability = \(\frac{4}{9}\). This adds up to
$$ P(win) = \frac{1}{2}\times 1 + \frac{1}{2}\times\frac{4}{9} = \frac{13}{18} $$
You can extend this recursively just for kicks. For example if every urn had two urns inside them and you had to fill them up with the balls to maximize you chances how would you do it? It is the same logic, one white ball in each and the last one should have all the remaining (shown in figure)
$$P(win) = \frac{1}{4}\times 1 + \frac{1}{4}\times 1 + \frac{1}{4}\times 1 + \frac{1}{4}\times\frac{2}{7} = 0.82$$
Here are 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
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
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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