The estimated jackpot for tonight’s Mega Millions state lottery (California, Georgia, Illinois, Maryland, Massachusetts, Michigan, New Jersey, New York, Ohio, Texas, Virginia, and Washington) drawing is up to almost $400 million.
The lottery has deservedly been called a stupidity tax, but how does this record jackpot change things? The odds of any specific ticket winning are 175.7 million to one. That gives a naive expected value of about $2.25 for each $1 ticket. Not bad, eh? Why not take out a short-term $175.7 million loan, purchase 175.7 million different tickets, and make a nice quick profit of $220 million or so? It’s easy money.
There are two reasons the $2.25 expected value is naive. The first is that the jackpot is only $400 million when it is paid out over 26 years, $15.4 million per year. The actual jackpot is about $239 million, which the winner may elect to receive as a lump sum. So a lottery ticket is still +EV, with an expected value of $1.36 or so, or perhaps $1.02 after taxes. (If you go ahead and take out that $175,700,000 short-term loan, you might expect to net about $3.5 million. Still easy money, right?)
The other reason is that there are likely to be multiple winners. If you expect to win but to share the jackpot with one other party, that cuts your expected value in half. With some simplifying presumptions, we could probably calculate the expected number of tickets identical to yours if we knew how many tickets had been sold, but that sounds like too much trouble. I think it’s safe to say that it’s large enough to make buying a random $1 ticket a -EV proposition in the end.
The stupidity tax is optional and I don’t contribute to it often, but I did make a two-dollar payment this afternoon. I figure that a lot of people choose numbers associated with birthdays or anniversaries, so I selected every number on both tickets from the range above 31. I figure I’m maximizing my EV by ensuring I wont be splitting the jackpot with any of those people. Is it enough to make my two-dollar bet a +EV proposition? Possibly, since there may also be non-jackpot payouts, but I doubt it.