Q:Three bullets are inserted into a gun which has six slots at random positions. The trigger is pulled, and no shot is fired. What is the probability a bullet will fire if the trigger is pulled once again? What is the probability of a fire in the 3rd pull if it is pulled twice and both are blanks?

A: Let us work through the cases. There are four possible configurations the revolver could be in. These are shown in the picture below.

There are a total of 6C3 configurations possible. Each of the first three configurations can happen in 6 ways, while the last configuration can happen in just 2 ways. Given that the first shot was a blank, the probability of firing a bullet in each of the configurations are the following

The total probability (as the above cases are mutually exclusive) is the weighted sum given as

$$P(Fire) = \frac{6}{20}\times\frac{1}{3} + \frac{6}{20}\times\frac{2}{3}\times 2 + \frac{2}{20}\times1 = \frac{3}{5}$$

If two pulls on the trigger goes blank only configuration 1 has a scenario where it could go blank on the third pull. I leave it to the reader to work out that case and probability.

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

Here are a few

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.

A: Let us work through the cases. There are four possible configurations the revolver could be in. These are shown in the picture below.

There are a total of 6C3 configurations possible. Each of the first three configurations can happen in 6 ways, while the last configuration can happen in just 2 ways. Given that the first shot was a blank, the probability of firing a bullet in each of the configurations are the following

The total probability (as the above cases are mutually exclusive) is the weighted sum given as

$$P(Fire) = \frac{6}{20}\times\frac{1}{3} + \frac{6}{20}\times\frac{2}{3}\times 2 + \frac{2}{20}\times1 = \frac{3}{5}$$

If two pulls on the trigger goes blank only configuration 1 has a scenario where it could go blank on the third pull. I leave it to the reader to work out that case and probability.

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

Here are a few

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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