Q: There are three coins, one of them is fair, while the other two are biased with probabilities of heads being \(\frac{1}{4}\) and \(\frac{3}{4}\) respectively. A coin is chosen at random. How many tosses are needed to have a greater than \(50\%\) probability that it is fair?

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

A: If \(n\) tosses are done of which \(t\) are tails then the probability of seeing them is given by

$$P(n,t | p=50\%) = \dbinom{n}{t}\frac{1}{2^{n-t}}\frac{1}{2^{t}} = \dbinom{n}{t}\frac{1}{2^{n}}$$

Likewise, the probability of it being the biased coin with \(p=25\%\) is

$$P(n,t | p=25\%) = \dbinom{n}{t}\frac{1}{4^{n-t}}\frac{3^{t}}{4^{t}} = \dbinom{n}{t}\frac{3^{t}}{4^{n}}$$

and the probability of it being the biased coin with \(p=75\%\) is

$$P(n,t | p=75\%) = \dbinom{n}{t}\frac{1}{4^{t}}\frac{3^{n-t}}{4^{n-t}} = \dbinom{n}{t}\frac{3^{n-t}}{4^{n}}$$

Putting things in a Bayesian perspective, we want to know \(P(p=50\%|n,t)\) which works out as

$$\frac{P(n,t|p=50\%)P(p=50\%)}{P(n,t|p=50\%)P(p=50\%) + P(n,t|p=25\%)P(p=25\%) + P(n,t|p=75\%)P(p=75\%)}$$

Simplifying the above equation with the values computed above, yields

$$P(p=50\%|n,t)=\frac{2^{n}}{2^{n} + 3^{t} + 3^{n-t}} \ge \frac{1}{2}$$

Note, the above equation holds only if \( 3^{t} + 3^{n-t} \le 2^{n}\). Do you see how? If \( 3^{t} + 3^{n-t} = 2^{n}\) then the above fraction would be equal to \(\frac{1}{2}\). From here onwards a brute force approach is needed as you have one equation with two unknowns. A simple piece of code that does this is shown below

The results show that at a minimum you would need 6 trials with 3 tails to conclude with greater than \(50\%\) probability that it is a fair coin. Note, when the tossing happens it may not work out this way, but you need at least 6.

If you are looking to learn the art of probability here are some good books to own.

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.

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

A: If \(n\) tosses are done of which \(t\) are tails then the probability of seeing them is given by

$$P(n,t | p=50\%) = \dbinom{n}{t}\frac{1}{2^{n-t}}\frac{1}{2^{t}} = \dbinom{n}{t}\frac{1}{2^{n}}$$

Likewise, the probability of it being the biased coin with \(p=25\%\) is

$$P(n,t | p=25\%) = \dbinom{n}{t}\frac{1}{4^{n-t}}\frac{3^{t}}{4^{t}} = \dbinom{n}{t}\frac{3^{t}}{4^{n}}$$

and the probability of it being the biased coin with \(p=75\%\) is

$$P(n,t | p=75\%) = \dbinom{n}{t}\frac{1}{4^{t}}\frac{3^{n-t}}{4^{n-t}} = \dbinom{n}{t}\frac{3^{n-t}}{4^{n}}$$

Putting things in a Bayesian perspective, we want to know \(P(p=50\%|n,t)\) which works out as

$$\frac{P(n,t|p=50\%)P(p=50\%)}{P(n,t|p=50\%)P(p=50\%) + P(n,t|p=25\%)P(p=25\%) + P(n,t|p=75\%)P(p=75\%)}$$

Simplifying the above equation with the values computed above, yields

$$P(p=50\%|n,t)=\frac{2^{n}}{2^{n} + 3^{t} + 3^{n-t}} \ge \frac{1}{2}$$

Note, the above equation holds only if \( 3^{t} + 3^{n-t} \le 2^{n}\). Do you see how? If \( 3^{t} + 3^{n-t} = 2^{n}\) then the above fraction would be equal to \(\frac{1}{2}\). From here onwards a brute force approach is needed as you have one equation with two unknowns. A simple piece of code that does this is shown below

The results show that at a minimum you would need 6 trials with 3 tails to conclude with greater than \(50\%\) probability that it is a fair coin. Note, when the tossing happens it may not work out this way, but you need at least 6.

If you are looking to learn the art of probability here are some good books to own.

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.

> A coin is chosen at random. How many tosses are needed to have a greater than probability that it is fair?

ReplyDelete0! Because given how they're specified to be biased, one can just use the http://en.wikipedia.org/wiki/Randomness_extractor#Von_Neumann_extractor procedure to turn them into producers of fair output! :)