More on the Lottery

LAGuy's post earlier today about why folks might choose one lottery over another has inspired me to do the math on something I've been pondering, i.e. leaving aside all other players and all lesser prizes, how much bigger does the Mega jackpot need to be for it to make more sense to play Mega than Lotto here in NY?

I'm in a lottery pool at work with 11 co-workers, just purely to be social. We each purchase $24 worth of tickets once every 12 weeks, taking turns. Here in NY we have Lotto and MegaMillions, and it's up to the person purchasing that week to decide what ratio of Lotto to Mega to purchase. So I thought I might want to know which was more rational to "favor" each week. To win Mega you must match five different numbers from 1 to 56 and one number from a separate pool of 1 to 46. To win Lotto you must match six different numbers from one pool of 1 to 59. Lotto tickets cost half of what Mega tickets do (i.e. they give you two for a buck).

First, the odds on winning Mega. Five numbers can be randomly selected from a field of 56 in 3,819,816 ways. (I used COMBIN(56,5) in Excel as a shortcut.) Adding in the final separate pool of 46 gives you 3,819,816 x 46 = 175,711,536. So your odds on any random Mega ticket winning the jackpot are one in 175.7 million. Meaning, of course, that it only makes sense to play Mega when the after-tax cash payout exceeds $176 million (leaving aside smaller payouts), but I'm not trying to dispute that the lottery is a tax on those who are bad at math. I'm just trying to figure out which one taxes us at a higher rate.

Doing the same formula COMBIN(59,6) for regular lotto gives you a one in 45,057,474 chance. Halve that, since the ticket effectively only costs $.50, and you get one in 22.5 million, approximately. Meaning that playing lotto only makes financial sense once the jackpot goes above $22.5 millionish.

Divide the probability of winning Mega by the probability of winning Lotto and you get 175.5/22.5 = 7.8. So as a rough working answer, it makes more sense to play Mega when it's at least 8 times as big as the Lotto jackpot. Of course, it never makes real sense to split your bets between the two games -- as we always do -- because one is always a better bet than the other. But as I said, this is mostly a social thing, and people gain utility from the anticipation/fantasizing/checking-the-numbers process for each drawing.

I'd bet there's a similar number for LAGuy's local drawings, but I'm not interested enough to do the research or math.