Q: You have a fair unbiased coin. How many times on average do you need to toss it to get \(n\) heads in a row.

Introduction to Probability Theory

A: This is another of those problems that can be best solved in a recursive manner. Assume that the expected number of throws to get \(n\) heads is \(x_{n}\). To get to \(x_{n+1}\) we would need to toss it one more time. But there is no guarantee that the next toss would result in a head. It would result in a head with probability \(\frac{1}{2}\) or a tail with the same probability. If it falls a tail, all our effort so far goes to naught and we have to start over with the added cost of having tossed once.

So, with probability \(\frac{1}{2}\) we toss it \(x_{n} + 1\) times and with probability \(\frac{1}{2}\) we toss it \(x_{n+1} + 1\). Phrasing this recursively gives us

$$x_{n+1} = x_{n} + \frac{1}{2}\times 1 + \frac{1}{2}(x_{n+1} + 1) $$

further simplifying to give

$$\fbox{ \(x_{n} = 2^{n+1} - 2\)}$$

If you are looking to learn probability here are some good 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

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.

Introduction to Probability Theory

A: This is another of those problems that can be best solved in a recursive manner. Assume that the expected number of throws to get \(n\) heads is \(x_{n}\). To get to \(x_{n+1}\) we would need to toss it one more time. But there is no guarantee that the next toss would result in a head. It would result in a head with probability \(\frac{1}{2}\) or a tail with the same probability. If it falls a tail, all our effort so far goes to naught and we have to start over with the added cost of having tossed once.

So, with probability \(\frac{1}{2}\) we toss it \(x_{n} + 1\) times and with probability \(\frac{1}{2}\) we toss it \(x_{n+1} + 1\). Phrasing this recursively gives us

$$x_{n+1} = x_{n} + \frac{1}{2}\times 1 + \frac{1}{2}(x_{n+1} + 1) $$

further simplifying to give

$$\fbox{ \(x_{n} = 2^{n+1} - 2\)}$$

If you are looking to learn probability here are some good 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

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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