09 July 2008

Lottery tickets with really bad odds

A CNN.com article talks about lottery tickets with zero probability of winning.

Why, you ask? Because some state lotteries continue selling the tickets for scratch-off games even after the top prize has been awarded. Therefore the odds stated on the ticket are, as of the time the ticket was purchased, incorrect.

But let's say that half the tickets for some game have already been sold, and the top prize not awarded -- then the tickets that are still out there have double the probability of winning that they did originally. You wouldn't see anybody complaining about that.

One way to fix this would be to have all the tickets be independent of each other, but drawn from the same distribution -- so instead of having one grand prize among the 100,000 tickets, each ticket independently has probability 0.00001 of being a grand prize ticket. But then there's a significant probability that there will be no grand prizes awarded, or that there would be two or more.

And some lottery websites actually state which prizes have already been awarded. So it might be possible for somebody to use this information to their advantage, by betting only in lotteries where a disproportionate number of prizes remain to be awarded. This is basically the same principle as card-counting in blackjack, where the player bets more when the cards in the deck are more favorable. I suspect, though, that this wouldn't work well because the house edge in lotteries is much higher than that in casinos.


Theo said...

In fact, students at MIT did succeed at cracking the MA lottery in exactly the way you describe in the last paragraph.

Michael Lugo said...


I didn't know that!

I'd heard of a related scheme, in lotteries where there's a "jackpot" that rolls over from week to week -- namely, when the jackpot is larger than the cost of buying all the possible combinations of numbers, do that.

The problems with this scheme are twofold:
- it requires buying millions of tickets, which is logistically difficult;
- one runs the risk of having to share the jackpot, cutting into the winnings.

Nick said...

It would probably just be simplest to clarify that the "odds" on a particular ticket are simply the odds at the time of printing and do not reflect tickets already sold.