Q: A group of 20 members is electing one of two candidates X and Y. You secretly know who will cast in favour of whom and know that X will win by 14 to 6 votes. What is the probability that X will be leading consistently when 12 votes are counted?

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A: As there are 20 votes in total, every path to the final state (14,6) can be thought of to be a state of counts of vote. In order to understand how many possible paths there are, you think of the problem as a combinatorics problem in the following way. When a given vote is counted it is either in favour of X or Y. This is shown in the figure below.

As there are 20 votes total, the number of ways by which the voting process will reach the final state can be ascertained by choosing 14 from 20, i.e. \(\dbinom{20}{14}\). The zone coloured blue in the figure is a \(6\times6\) grid. This is a good time to introduce the concept of "Catalan numbers". The Catalan numbers is a handy tool that can be applied to several combinatorial problems including this one. The number of paths in an \(n\times n\) grid that either touch the diagonal or are above it is given by the Catalan number.

$$

C_{n} = \frac{(2n)!}{(n+1)!n!}

$$

For the \(6\times6\) grid these paths, as computed by the Catalan number are the cases favourable for a consistent lead. So the sort probability works out to be

$$

\frac{\Big(\frac{12!}{7!6!}\Big)}{20\choose14}

$$

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

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.

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (And Everone Else!)

A: As there are 20 votes in total, every path to the final state (14,6) can be thought of to be a state of counts of vote. In order to understand how many possible paths there are, you think of the problem as a combinatorics problem in the following way. When a given vote is counted it is either in favour of X or Y. This is shown in the figure below.

As there are 20 votes total, the number of ways by which the voting process will reach the final state can be ascertained by choosing 14 from 20, i.e. \(\dbinom{20}{14}\). The zone coloured blue in the figure is a \(6\times6\) grid. This is a good time to introduce the concept of "Catalan numbers". The Catalan numbers is a handy tool that can be applied to several combinatorial problems including this one. The number of paths in an \(n\times n\) grid that either touch the diagonal or are above it is given by the Catalan number.

$$

C_{n} = \frac{(2n)!}{(n+1)!n!}

$$

For the \(6\times6\) grid these paths, as computed by the Catalan number are the cases favourable for a consistent lead. So the sort probability works out to be

$$

\frac{\Big(\frac{12!}{7!6!}\Big)}{20\choose14}

$$

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

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.

Can you please clarify what you mean by "What is the probability that X will be leading consistently when 12 votes are counted?"

ReplyDeleteDoes it mean that after every vote that is counted till 12, X should be leading?

Unfortuntly, your answer is wrong... The number of paths in which X consistently leads Y relates to Dyck words (search the web for the number D(m,n), which counts the number of path in an n by m rectangle which does not go above the diagonal of the left n \by n sub-rectangle). Using this value (that relates to Catalan numbers), you can prove that the probability is 9/15.

ReplyDeleteI think you are right. Thanks much for pointing that out.

ReplyDelete