Q: Four teams are selling selling lemonade to customers walking out of a store. A total of 10 customers walk out of the store and randomly pick a team to buy lemonade from. What is the probability that all teams get at least one customer?

Probability Theory: The Logic of Science

A: For simplicity, assume that the customers are walking out in a line and pick a team to buy lemonade from as shown in figure below.

The probability that a given team would get at least one customer is one minus the probability they get none. For a given customer and a given team, there is a \(\frac{4 - 1}{4} = \frac{3}{4}\) probability that they will not connect. The probability that none of the 10 customers would pick a given team is \(\big(\frac{3}{4})^{10}\).

Moving on, the probability that exactly one team would be without customers is \({4 \choose 1} \times\big(\frac{3}{4})^{10}\), likewise the probability that exactly two and three teams would be without customers are \({4 \choose 2} \times\big(\frac{3}{4})^{10}\) and \({4 \choose 3} \times\big(\frac{3}{4})^{10}\) respectively. Thus the probability that all teams get at least one customer is one minus the sum of all the above probabilities

$$

P(\text{at least 1 customer}) = 1 - \big(\frac{3}{4})^{10}\times(4 + 6 + 4) \approx 21\%

$$

This problem is also a good candidate to try out some simulation and learn some R programming along the way. The following code simulates the above process and creates the same result.

Discovering Statistics Using R

If you are looking to learn probability, some good books to own are shown below

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

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.

Probability Theory: The Logic of Science

A: For simplicity, assume that the customers are walking out in a line and pick a team to buy lemonade from as shown in figure below.

The probability that a given team would get at least one customer is one minus the probability they get none. For a given customer and a given team, there is a \(\frac{4 - 1}{4} = \frac{3}{4}\) probability that they will not connect. The probability that none of the 10 customers would pick a given team is \(\big(\frac{3}{4})^{10}\).

Moving on, the probability that exactly one team would be without customers is \({4 \choose 1} \times\big(\frac{3}{4})^{10}\), likewise the probability that exactly two and three teams would be without customers are \({4 \choose 2} \times\big(\frac{3}{4})^{10}\) and \({4 \choose 3} \times\big(\frac{3}{4})^{10}\) respectively. Thus the probability that all teams get at least one customer is one minus the sum of all the above probabilities

$$

P(\text{at least 1 customer}) = 1 - \big(\frac{3}{4})^{10}\times(4 + 6 + 4) \approx 21\%

$$

This problem is also a good candidate to try out some simulation and learn some R programming along the way. The following code simulates the above process and creates the same result.

Discovering Statistics Using R

If you are looking to learn probability, some good books to own are shown below

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

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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