Winning the Lottery, Twice!

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You might have heard this piece of news often "XYZ wins lottery for a second time". Such news, like here, leaves the reader wondering how someone can get that lucky. Needless to say, the winner is on cloud nine knowing s/he has won, not once but twice! But are these events truly rare? We are conditioned to believe that winning a lottery in itself is a rare event let alone winning it twice. Lets explore.

Probability Theory: The Logic of Science

To begin understanding the rarity of the above event, lets revisit the Birthday Problem. The summary of the problem is we need an astonishingly small number of people in a room to have \(>50\%\) probability that two people would have the same birthday, about 23. To generalize, there are 365 days in a year. If there are 2 people in a room, the probability that both have different birthdays are \(\frac{364}{365}\). If there are 3 its \(\frac{364}{365}\times\frac{363}{365}\) and so on. If you assume there are \(N\) days in a year and there are \(r\) people we can compute the probability that they all have different birthdays as
$$
P(\text{different birthday}) = \frac{N}{N}\times\frac{N-1}{N}\times\frac{N-2}{N}\ldots\frac{N-r-1}{N}
$$
Implying the probability that at least two have the same birthday is \(1 - P(\text{different birthday})\)

These results can be extended to the double-win lottery phenomena as well. \(N\) can be mapped to the possible numbers in a lottery, \(r\) could be the number of lottery players. Let us make some simplifying assumptions.

• The winners of a lottery would continue buying tickets and most lottery buyers who don't win continue to buy lotteries, and assume all lotteries are drawn weekly.
• The lottery draw is done once a week.
• It is the same set of \(r\) people who are buying every week and the winner is one of the \(r\)
• Everybody buys one ticket which has a distinct number.

To compute the probability that in the second week the same person does not win, two things need to happen.

1. The winning number has be one of the \(r\), this happens with probability \(\frac{r}{N}\)
2. The winner in the first round should not be the winner in the second round. This happens with probability \(\frac{r-1}{r}\).

Thus, the overall probability of not having a double winner in the second week is \(\frac{r}{N}\times\frac{r}{N}\times \frac{r-1}{r} = \frac{r(r-1)}{N^2}\). For the 3rd week, by applying a similar logic, we get the probability that all three are distinct winners as

$$
P(\text{all 3 distinct}) = \frac{r(r-1)(r-2)}{N^3}
$$

Let us examine what happens over a decade. That is a total of 520 draws. Let us also put some real numbers behind this. Assume there are 10 million lottery players and there are 175 million numbers to choose from (these are rough estimates from Powerball numbers). The value of \(P(\text{all distinct})\) works out as
$$
\frac{10^{6}\times(10^{6} - 1)\times(10^{6} - 2)\ldots\times (10^{6} - 520)}{(175\times 10^{6})^{520}}
$$
The above expression is in-computable by any machine we know of, however we can easily find a maximum possible value for this expression. The numerator is clearly less that \(10^{6\times 520}\) and the denominator can be factored out as \(175^{6\times 520} \times 10^{6\times 520}\). This gives a maximum bound for the expression as
$$
\frac{1}{175^{520}} \approx 0
$$
This implies, the probability of there being a double winner over a decade, under the prevailing assumptions, is almost \(100\%\)!
Why then have we not heard of double winners in Powerball jackpots? The most likely reason is the winners of big lottery draws likely don't bother coming back and buying more. Also, from the point of view of the buyer the probability of a double win IS astronomically small, however the probability that there would be a double winner somewhere or the other (under the prevailing assumptions) is quite the opposite, it is a mathematical certainty!

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
Overall an excellent book to learn probability, well recommended for undergrads and graduate students

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.

Introduction to Probability, 2nd Edition
A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.

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 and even 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
This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"

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 Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

Probability and Statistics (4th Edition)This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good