Whether you can beat the lottery falls into the category of questions that interest mainly men badly in need of a shower and a shave. You don't need a degree in probability theory to know that the quick answer is "no."
And yet: I've wondered about whether there might be exceptions to this rule ever sine reading Jonah Lehrer's (yes, that Jonah Lehrer; this is not the point to rehash the whole Lehrer debate) story about folks who've cracked high-dollar scratch-off lottery tickets. One person mentioned in the story is a Texas mathematician who has won several jackpots. Lehrer followed up with a post about GS Investment Strategies, which used a quirk in the Massachusetts lottery rules to cash in on purchases of hundreds of thousands of dollars of tickets.
It seems conceivable that there may be similar opportunities still out there. So let's pose the question another way: are there times when the payouts in a lottery will exceed the total cost of the tickets, and how do you find them?
The answer to the first part is yes: when most of the tickets have been sold, and the grand prizes still haven't been claimed. Now the second part seems harder. But if you investigate New York's scratch-off games, there is one key piece of data published every week that helps you answer it: the number of unclaimed top prizes in each lottery game.
This list comes out every Monday, and the key here is that it includes not just the number of top jackpots still out there, but the number of second prizes as well. This last part is key, because there are generally a lot more of those second prizes. What you can do with that information is estimate what share of the tickets have already been sold.
Last week I started looking at one of the games, the $10 "Monopoly" tickets. The top prize in that game is a $2 million payout; second prize is a mere $10,000. Normally the odds of winning the second prize are 10.5x higher than the top prize -- I would assume there are two prize tickets and 21 second prize tickets in the print run.
Right now, though, there are still two unclaimed jackpot tickets, and just three second prize tickets left. With 6/7 of the second prize tickets claimed, it's likely the only about 1/7 of the tickets are left. If that's the case the chances of a jackpot rise from by 7x, from 1 in 2,642,000 to about 1 in 377,429. Unfortunately the payout on the rest of the tickets is fairly meager. Even at those odds, you will make only $8.93 on the average $10 ticket (before taxes!)
Last night, though, I saw another candidate pop up, a $5 game called the "The Color of Money." This game has a surprisinggly high payoff to start for the non-jackpot tickets. On average you can expect to get $3.07 in smaller prizes for each $5 ticket -- already a surprisingly good payout for a $5 game.
Now here's the good part: right now there's just one $5,000 second prize still to be claimed, and one grand prize of a million dollars doled out at $50,000 a year/20 years. If -- if -- the second prizes have been found at the expected level of random chance, then more than 90% of the tickets are gone. And the odds of getting the grand prize have gone from 1 in 2,595,600 to 1 in 247,200.
Exactly how much you value the payout will depend on the discount rate you apply. At the target inflation rate of 2%, the prize is worth $817,572. if you use the thirty year treasury yield, it's $759,942. At that value your expected return on this is (coincidentally) another $3.07, for a total of $6.14 in returns for each ticket.
You can do your own calculations here on the risk-adjusted returns. Buying up all the tickets would be no mean feat. More important: if the second prizes have been won at a little higher than the expected rate, and there are, say, 500,000 tickets out there instead of 247,200, you are hosed. Nonetheless, overall it seems that it may be possible under some circumstances to take advantage of the limited indication of how many prizes remain to be won.
For the record, yes, I bought two Color of Money scratch-off tickets. I did not win anything, making the odds now ever so slightly better for everyone else.