Chances of Duplicate Lottery Tickets - Special Edition

Introduction
We've often wondered what the likelihood of different lottery players having the same combinations in a single drawing actually was. Looking for an answer, we searched the internet, but could not find it. So, we decided to conduct our own research. This was performed by studying the classic Identical Birthday Problem, identifying the underlying mathematical formula, applying this formula to individual lotteries, and summarizing our results.

Birthday Problem
How often have you been in a group of people and discovered that two of you shared the same birthday? Was this purely coincidence, a random event? Or, was it in fact highly likely?

The classic form of the Birthday Problem, which is familiar to most everyone, quantifies the chances of two people sharing the same birthday. Given a probability of certainty, the Birthday Problem solution calculates the size of smallest group necessary to meet that probability.

Thus, when in a group of 23 (22.5 actually) people, you can be 50% certain that two or more of you share the same birthday. To be 99.9% sure, you need a group of 71 people!

The Mathematics
The formula behind this solution is fairly simple, and in terms of Excel is written as:

SQRT(2*PopulationSize*LN(1/(1-Probability)))

We solve the Birthday Problem by substituting the PopulationSize with 365 days and the Probability of 0.50 or 0.999 to acheive the answers above.

Calculating the Chances of Duplicate Lottery Tickets
We applied the formula above to the various lottery games that we cover and produced the Chance of Duplicates Table below.

The first column identifies the lottery Game. Next is the Population (total number of possible combinations) for that game. The 3rd and 4th columns are our results. Column A identifies the minimum number of tickets that must be issued in order to be 50% sure that there at least one duplicate. Column B is similar, but identifies the minimum number of tickets that must be issued in order to be 99.9% sure that there at least one duplicate.

Chance of Duplicates
Game	Population	Col A 50% Sure	Col B 99.9% Sure
Powerball	146,107,962	14,232.0	44,928.3
Powerball (Jan 09)	195,249,054	16,452.1	51,937.1
Mega Millions	175,711,536	15,607.3	49,270.1
Lotto 649	13,983,816	4,402.9	13,899.4
Super 7	62,891,499	9,377.4	29,476.7
Super Lotto Plus	41,416,353	7,577.3	23,920.5
Hot Lotto	10,939,383	3,894.3	12,293.6
EuroMillions	76,275,360	10,283.0	32,462.0
Irish Lotto	8,145,060	3,360.3	10,607.9
UK Lotto	13,983,816	4,402.9	13,899.4
Thunderball	3,895,584	2,323.9	7,336.2
Birthdays	365	22.5	71.0

The Chances
As shown, the chances that duplicate lottery tickets will be sold are very likely. For example, there is a 50% chance that duplicate Powerball tickets will be issued when only 14,232 tickets are sold. And, you can be 99.9% sure that there are duplicates when only 45,000 tickets are sold. When Powerball changes format in January 2009, the total number of possible combinations will increase by over 49 million. Even so, you can be 99.9% sure that there will be duplicates when only 52,000 tickets are sold. By reading the table above, you can determine the likelihood of duplicate tickets in your favorite lottery, whether it be: Powerball, Mega Millions, Super Lotto Plus, Hot Lotto, EuroMillions, Irish Lotto, UK Lotto, or Thunderball.

Summary
This study identifies the number of tickets that must be sold in order to be 50% and 99.9% mathematically certain that one or more people will have duplicate combinations. While this information is not sufficient to estimate the total number of duplicate tickets, it provides a guideline to understand the chances. If you live in a small town of around 50,000 people, and everyone buys 1 lottery ticket, don't be surprised if you discover that someone else has the same combination as yours.

JL .........


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