# The value of a Megamillions lottery ticket

I’m fascinated by these new fangled national lotteries.   In most gambles, whether the casino or lottery variety, the math never works out in your favor.   That’s, after all, how these businesses make their money.   But, in Megamillions the “house” (in this case, the state government) takes a portion of profit from each ticket, rather than gambling against us collectively.   If no one wins, the pot gets bigger.   In this sense it operates in some way like casino poker, or horse racing in Hong Kong.   The “house” is out of it, so you don’t have to worry — as much — about the game being rigged.   You’re competing against everyone else.

I find the math around this interesting.   I’ll start with the premise that the value of a gamble is equal to the pot multiplied by the chance of winning.  If the value of the gamble supplied by this equation is more than the cost of the buy-in, then the gamble “makes sense”.   It would make sense to pay a dollar for a 50% chance of winning four dollars.   It would not make sense to pay two dollars for a 50% of winning three.

Note: I find this math a bit simplistic for serious consideration.   I’m just thinking out loud.   No assessment of gambling is complete without some mechanism for gauging the relative value of money.   In other words, a dollar for a millionaire is not worth the same amount as a dollar for a guy who needs it to pay his rent.  Also, there should be some mechanism for computing the elasticity of the value of money.   If I have \$10 after paying rent, then it might make sense to spend \$1 on a twinkie.   But spending \$5 on twinkies probably doesn’t make sense.  And spending \$20 on twinkies definitely doesn’t.    My simple equation doesn’t take that into account, either.   Basically this is a complicated way to say that I am in no way encouraging anyone reading this to play the lottery.   I just find the math interesting and wanted to share.

That being said, here are my computations for the value of a Megamillions lottery ticket.

Winning Combination Winnings Chance of Winning Value = Winnings x Chance of Winning
All five white + the yellow The jackpot 1:258,890,850 Depends on jackpot
All five white \$1 million 1:18,492,204 \$0.01
Four white + the yellow \$5 thousand \$739,688 \$0.01
Four white \$500 1:52,835 \$0.01
Three white + the yellow \$50 1:10,720 \$0.00
Three white \$5 1:760 \$0.01
Two white + the yellow \$5 1:473 \$0.01
One white + the yellow \$2 1:56 \$0.04
One white \$1 1:21 \$0.05

I find it interesting that the value of a winning increases slightly on the very low end. I imagine that’s because a lot of people who win these small pots put the money right back into buying more tickets, which means the house ends up with that money anyway.

So, other than the jackpot, the value of all the other winnings put together is just 18 cents. Since a ticket costs one dollar, that’s not a very good deal. The jackpot fluctuates wildly. So the real question is: at what point does the jackpot make the whole thing worth it. The equation in question would be:

( X / 258,890,850 ) + 0.18 ≥ 1

… where X is the value of the jackpot.

∴ X ≥ ( 1 – 0.18 ) * 258,890,850 = \$212,290,497

In theory, then (and I’d remind you of the two caveats above), it makes sense to buy a Megamillions lottery ticket when the pot goes above 212 million or so.