Q: You participate in an raffle. You have bought a certain number of raffle tickets and there are several prizes to choose from. Assuming you don't care amongst the prizes, do you put all your tickets in one box or distribute them across all the boxes?

A: Several offices have these kind of raffles internally during holiday seasons (Christmas) and I thought it would be worthwhile to formalize it with some proofs and simulations. Assume you have bought "r" tickets and that there are "N" total tickets outside of the "r" you bought. Also assume that there are "m" prizes to be won. You are faced with two broad options.

1) Put them all in one box.

2) Put them evenly in all the boxes. That is, each box gets r/m raffle tickets.

An example of the two options is shown below.

If you assume that all the other raffle tickets are placed evenly, then the probability of winning a prize, should you go with option 1 is simply

If you go with option 2: to compute the probability of at least one win, first compute the probability that you will lose all "m" bets, this works out to

Implying the probability of at least one win to be one minus the above, yielding

It now comes down to comparing one vs the other. For those interested, you could try to prove that option 2 is always greater than option 1, so one must always spread your raffle tickets out.

If you are looking to buy some books in probability here are 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

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: Several offices have these kind of raffles internally during holiday seasons (Christmas) and I thought it would be worthwhile to formalize it with some proofs and simulations. Assume you have bought "r" tickets and that there are "N" total tickets outside of the "r" you bought. Also assume that there are "m" prizes to be won. You are faced with two broad options.

1) Put them all in one box.

2) Put them evenly in all the boxes. That is, each box gets r/m raffle tickets.

An example of the two options is shown below.

If you assume that all the other raffle tickets are placed evenly, then the probability of winning a prize, should you go with option 1 is simply

If you go with option 2: to compute the probability of at least one win, first compute the probability that you will lose all "m" bets, this works out to

Implying the probability of at least one win to be one minus the above, yielding

It now comes down to comparing one vs the other. For those interested, you could try to prove that option 2 is always greater than option 1, so one must always spread your raffle tickets out.

If you are looking to buy some books in probability here are 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

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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