# bitguru blog

## Lottery Denouement

Posted by bitguru on March 8, 2007

The jackpot prize pool for Tuesday’s Mega Millions lottery drawing turned out to be $233 million, corresponding to an advertised jackpot of$390 million. The unclaimed jackpot prize pool for the March 2nd drawing was approximately $164.3 million. The jackpot prize pool is 31.8% of sales so, at a doller per a ticket, 216 million tickets must have been sold for the March 6th drawing to account for the$68.7 million difference. That’s a lot of tickets. Presuming that the numbers on those tickets were chosen uniformly, we’d expect 1.23 of those 216 million tickets to exactly match your ticket. Even if your crazy time-travelling uncle told you the winning numbers in advance, you would expect to be splitting the jackpot two or three ways.

In fact, two jackpot-winning tickets were sold, one in Georgia and one in New Jersey. Each winner will choose between a lump sum of $116.5 million or a$195 million annuity. The winning numbers were 16, 22, 29, 39, 42, and Mega Ball 20. As usual, my strategy of choosing only numbers above 31 didn’t pan out.

When I last checked a couple of years ago, not a single Powerball or Mega Millions drawing had ever resulted in all balls above 31. I expected such results to be rare, but not quite that rare. Just how often should we expect all of the winning numbers to be greater than 31?

A Mega Millions drawing has 5 white balls chosen from 1–52 and one yellow ball chosen from 1–46. The probability of all balls above 31 is $\frac{ { 21 \choose 5 }{ 15 \choose 1 } }{ { 52 \choose 5 }{ 46 \choose 1 } } = \frac{ ( 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 ) 15 }{ ( 52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 ) 46 } \approx$ 0.00255. There were fewer white balls before June, 2005, so the probability would have been higher then. But using the lower figure, we would expect two or three of the 1022 drawings since 1996 (including The Big Game drawings) to qualify. There have in fact been two:

A Powerball drawing has 5 white balls chosen from 1–55 and one red ball chosen from 1–42. The probability of all balls above 31 is $\frac{ { 24 \choose 5 }{ 11 \choose 1 } }{ { 55 \choose 5 }{ 42 \choose 1 } } = \frac{ ( 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 ) 11 }{ ( 55 \cdot 54 \cdot 53 \cdot 52 \cdot 51 ) 42 } \approx$ 0.00320. There were fewer white balls before August, 2005, so the probability would have been higher then. But using the lower figure, we would expect about five of the 1553 drawings since 1992 to qualify, but there have been only two:

Three times in the last four months after a drought from 1992–2005: I’m not sure what to make of that.

By the way, it was more work than I expected to gather the historical lottery data. I figured a quick web search would do the trick, but most of the data sets I found had been munged to make them useless for anything except proprietary lottery-prediction software. Others data sets were incomplete.

The most useful data set was this one from the Powerball web site, though it goes back only as far as 1997. I had to import the older drawings a few months at a time. For Mega Millions (and its predecessor, The Big Game) I could import drawings a year at a time via this page.

If anyone cares, I’m making the data sets I compiled available in plain text to anyone who wants them. The Powerball data set contains 1553 drawings from 22 March 1992 through 7 March 2007. The Mega Millions/The Big Game data set contains 1022 drawings from 6 September 1996 through 6 March 2007. Use them at your own risk. Some or all of the data may be inaccurate. I won’t be updating them, so they soon will be out of date.

The format of the two data sets isn’t quite the same, but it’s close. You should ignore any line that is entirely whitespace or that begins with the # character. The remaining lines may be parsed in Perl or Java4 with split("[^0-9]+"). That will yield an array of values which can be converted to integers, the first nine of which will be: month, day, year, ball1, ball2, ball3, ball4, ball5, special ball.