Math and economics of Powerball

It’s fun to think about winning enormous lottery jackpot ($1.3 billion annuity, $806 million cash value), and that alone is a fine reason to buy some Powerball tickets for Wednesday’s drawing.  But does it make sense as a gamble?  Here are some answers.

The main answer is “no”, it is a bad gamble because the expected value of a ticket is much lower than the $2 cost of a ticket.  You can check this yourself by hand if you keep in mind that the cash value is $806 million, each ticket costs $2, and they plan to sell at least 350 million tickets for the upcoming Wednesday drawing.  (That is no surprise, since this past Friday saw about 138 million tickets sold and Saturday had about 200 million tickets sold, according to this Yahoo news story.  Surely even more tickets will be sold for Wednesday’s drawing.)  Because the ticket sales are so large, you risk splitting the jackpot, which makes it a worse bet.

Aaron Abrams and I wrote a paper where we ran the numbers on lottery drawings.  In it, we developed a rule of thumb for checking the expected value of tickets in lotteries with big ticket sales (see subsection 7.2).   It is easy to see that this coming Wednesday’s Powerball drawing meets these criteria, because they are selling more than 350 million tickets and the jackpot is less than $995 million cash value.  This is a little easier than grinding out the expected value by hand, and  of course the rule of thumb also says that the expected value is negative.

In our paper we developed a model of lottery sales and predicted that Powerball and Mega Millions would never be a “good bet”, meaning would always have negative expected value.  So it is no surprise that this particular drawing has negative expected value.

It does happen sometimes that an occasional lottery drawing has positive expected value, just not Powerball and Mega Millions.  But even then, economics (mean-variance portfolio theory) tells you that buying lottery tickets is not a good investment.  Read the paper for details!

All that said, I’m going to buy a few tickets for Wednesday’s Powerball drawing.  I don’t expect to win anything, but it’s fun to think about it.

Reporters/journalists: If you are looking for a mathematician to talk about the lottery, I suggest Aaron Abrams, Curt Bennett, or Paul Jung.