The lottery is something that’s been around for the longest time. It used to be known as the “numbers” racket and was administered on a local level until states got smart and realized that it could be a good way to boost their coffers. Now, we have two national lotteries which have displaced the big state-run ones. Sometimes, due to luck, or lack thereof, those can get large. Megamillions and Powerball: what are the differences and what does it matter to the person who only plays in an office pool when the prize gets large enough?
- All states except for: [AL, CO, NV, UT, WY] + FL (starting May ’13). Draw 56 balls, choose 5. Then 46 balls, choose 1. Odds: 56C5*46 = 1 in 175.7 million
- All states except for: [AL, CO, NV, UT, WY] + CA (starting April ’13). Draw 59 balls, choose 5. Then 40 balls, choose 1. Odds 59C5*40 = 1 in 175.2 million
So, the odds are essentially the same. That doesn’t mean much on its own. Most people only play when the jackpot gets big, because you can increase the odds of winning by pooling with others (20 coworkers or friends contribute the same amount for the same share in a pot) in the big draws. The theory goes, even when dividing by 20, it’s still a big haul. However, in the big draws, it’s no guarantee you or your group wins on its own. So, the point of the post is to suggest the best situations for an office pool.
The key to lotteries is that you’re playing for “dead” money against “new” money. The lotteries declare that of the new money, 29% goes towards the current jackpot, 21% towards side jackpots (non-jackpot combinations), and 50% back to the states — education budget, lottery admin, etc. So, you don’t want to be playing against the new money because most of it gets siphoned off. If the dead money, “jackpot,” is sufficiently low, it’s not a great play. Sure, you can still win, but your likelihood of doing so is low and the take isn’t all that great.
This dead money concept is essentially the same concept as poker. If you’re in late position with a random/garbage hand and there are antes in the pot, you should be more likely to open (raise), than if there’s just the blinds’ money. In mega-lottos, you can be late to act, since there are projections on the next jackpot and you don’t have to do anything until 15 minutes prior to the draw. There occurs a point where you have maximal expected value. If most people are simply playing an office pool with their eyes on the grand prize, it’s too late. If you don’t want to wait until the last second, you can always predict human behavior.
To figure out the expected value per dollar value of jackpot, I turn to historical data run from the past five years. If we assume a jackpot plowback (the amount the jackpot is expected to increase between drawings) of $100mm on a 29% rate, that would imply $344mm tickets were bought, which have an equal chance of winning. The expected value of a ticket continues to rise at a linear rate and as jackpots get large, that falls apart and might even reverse itself. The methodology is to take the odds of winning times expected cash jackpots divided by the number of (expected value) other winners. If too many people play, they’re going to split up the dead money. A quadratic regression seemed to provide the best fit.
Conclusion: As the jackpot rises, your expected value per ticket rises. However, once it gets too large, there are many other players and the odds of a split ticket increase. The best time to play in an office pool is when the cash (not top-line annuity) jackpot is between 150 and 250 million. There’s really no situation where the expected value ever reaches north of a $1 on a pre-tax basis, so it’s never correct to buy every single combination of tickets.
Afternote: A random, computer-generated hand is probably better than anything else you might encounter, such as a fortune cookie, significant birthdays or numbers of retired Yankee jerseys, which increases your chance of running into co-winners, since the numbers tend to be clustered on the low side and someone could have the same exact numbers. Again, the odds of winning are independent; you don’t want to be sharing if your lucky numbers are called. A computer-generated draw using numbers over 31 would probably be the most optimal play, though probably not worth the extra effort.