Wednesday, January 13, 2016

Printing Every Powerball Lottery Number

I think gambling is fascinating. I think it's tempting and I abstain from it. I think it's morally wrong. Value comes from work. You don't get something for nothing in life (there's always a loss to someone).

Tonight the Powerball will play for a world record \$1.5 billion. Billion! That's insane. The odds are also insane. Powerball is played by choosing 5 of 69 numbers and then choosing 1 of 26 "powerballs." How many ways can the Powerball numbers be chosen? That's easy:

\binom{69}{5} \binom{26}{1}

which is:

\frac{69!}{5!(69-5)!} \cdot \frac{26!}{1!(26-1)!}

which is 292,201,338. That's a very big number. I know 1.5 billion is 5 times bigger but it's still a big number. Well let's consider the title scenario of printing out every single Powerball combination possible. Just what would that look like?

When I made a quick test print of different font sizes from 1.5 to 5.5 in half-point increments, I thought 3.5-point Courier New made for a nice size. That's a little more than 1 mm in text height. Making some adjustments to line spacing, page margins, and using 15 columns allowed me to put 2970 Powerball combos on a sheet of paper. If we print double sided we get 5940 combos per sheet of paper (enough combos for 57 years of twice-weekly playing). That means we'll need 49193 sheets of paper or 99 reams of paper! That weighs in at 492 lb. After toner is applied it's sure to weigh more than a quarter of a ton (sorry planet earth). Between me and the paper, I'm pushing the limit of what my Toyota Yaris is rated to carry. (Despite my wife's objection to having 10 cases of useless paper sitting around our house, I think it would be cool to have a copy.)

Walter Hickey wrote an excellent analysis for Business Insider of the expected value of a lottery ticket (written in Sept. 2013). He takes into account the possibility of sharing the jackpot (because more people buy tickets as the jackpot goes up in value) and the loss in value due to taking a lump sum. He does not factor in taxes, though. The short answer is that after taxes, it's never worth playing the lottery. This is no surprise.

The jackpot didn't get to \$1.5 billion by magic. It gets that high because people lose! That \$1.5 billion represents hundreds of millions of \$2 tickets. More tickets than there are combos! The odds truly are terrible. People will often say, "You have to play to win." I would say that the odds are so bad that not playing is as good as winning \$2 (or more!) twice a week for the rest of your life.

Here's a fun simulator (play as much as you want!):