Obama '666' lottery... debunked!

A friend of mine brought this story to my attention:
http://www.snopes.com/politics/obama/lottery.asp

The short version?  The day after Obama's 2008 election, the winning numbers of the Illinois Pick-3 state lottery was 666.  So naturally, the conspiracy theorists freaked out over it. After all, the chances of this are 1/1000, right?  Well, not quite.

Being the nerd and the asshole that I am, I can't pass up the opportunity to both work on a fun math puzzle and debunk a conspiracy theorist.  So here we go.

Let L be the number of "Pick-3" daily lottery drawings that happen in the US on a given day.  Most states have a daily Pick-3 lottery: Colorado, NJ, Arizona, Wisconsin, California, to name but a few.  In fact, Illinois has the drawing TWICE daily, as does Ohio, Missouri, Maryland, Pennsylvania, Florida, New Hampshire, Kentucky, Virginia, Maine, New York, and probably more.  Texas does their Pick-3 only six days a week, but FOUR times daily.  A handy source for these is the lottery links page on usa[.]gov.  In any case, the probability that 666 will NOT show up in any of these L lottery drawings is (999/1000)^L.

Now let n be the number of different days that we can associate as being somehow specially related to President Obama. This is not limited to "the day after Obamas [sic] election", but also the very day of the 2008 election, the day before, the day of the 2004 election, the days before and after this, the day he was sworn in, the one-year anniversaries of these days, and so on.  The probability of NOT getting 666 on one of these "significant" days is therefore: ((999/1000)^L)^n.

Therefore, the probability that you DO get '666' on at least one of these lottery drawings on one of these days is 1 - ((999/1000)^L)^n.

For the sake of the argument, let's say that L=40 (though it's probably higher).  As for special days, I only just named 14 off the top of my head.  With these numbers, you get a roughly 43% chance of '666' showing up.  To have a greater than 50% chance of '666' showing up in one of these lotteries on a 'special' day, all you need to have is n>17.  And again, this is assuming that L=40, when it's probably higher.  We're also ignoring Pick-4 drawings that could begin or end with '666'.  Heck, we're ignoring all other sorts of events in which the number "666" might randomly show up in some form or another: stock prices, budget costs, etc.

QED
(though if you find a flaw in my math, please do email me and I'll correct it)

The moral of the story: numerical coincidences like this aren't so amazing when you take into account the full number of statistical trials you can have.  In this case, that means looking at more than just one lottery and looking at more than just one day deemed to be somehow significant.