## Wednesday, March 28, 2012

### Mega Millions, Probability, and You

If you haven't heard, this Friday's Mega Millions lottery jackpot, which already has an estimated pay out of \$500 million, will be the largest lottery prize ever won (assuming somebody hits all six numbers).

It's hard for me to fathom that much money. That's half a billion dollars, just for matching numbers. Incredible.

Now the odds for winning that prize are a little more than 1 in 175 million. This got me thinking, if the jackpot has a larger number than what the odds are, shouldn't every person have a reasonable positive expectation if they played the lottery? But, you can't really ever expect positive returns from a lottery, right?

As math is a fuzzy subject for me, I deferred to my buddy the math whiz (he wrote a post for me on the NBA lockout a few months ago). Here's what he had to say.

"Hey Bry,

Great question.  Some of the math of probability isn't super fresh in my mind (though I'd say it's one of my favorite areas of mathematics), but I'll offer what I can.  First, I would say that what you said isn't wrong, but the situation is more complex.

It's important to remember that there are both helpful and harmful nuances to the system that affect the expected return on a lottery ticket.  On the helpful side, you have opportunities to make money without hitting the jackpot (if you get x, y, or z numbers right).  On the "harmful" side (as it relates to expected return), you could win the jackpot but end up having to split it with another winner, or two other winners, etc.

Let's pretend that those nuances weren't there.  (You may know all of what follows in this paragraph, but I'll just cover the bases.)  If you were to buy 175,000,001 lottery tickets for \$1 each, you'd of course spend \$175,000,001.  Odds are, 175 million of those tickets would not hit the jackpot and thus would give you no money.  The remaining ticket would hit the jackpot, which we'll pretend is \$350 million (not far from the actual current situation, right?).  That means your profit from buying all those tickets would be \$175 million (rounded up \$1).

The thing is, you're obviously not going to buy that many tickets, and that's where expected return comes in.  To find the expected return on a ticket, I believe you would divide the profit by the cost in the above scenario (at least in this case, because the number of tickets purchased is equal to the \$ spent).  So, yes, you'd have a positive expected return of \$1 for every ticket purchased (ie, a *profit* of \$1 for every ticket purchased).  However, the concept of expected return isn't very helpful on an all-or-nothing investment with incredibly long odds.  The reality is that there is a very high chance of you winning absolutely nothing, and that only changes if you buy an incredible number of tickets.

Then, the math would be further complicated by the nuances mentioned above."

What do you think? Will you be trying to win Friday's prize?  Let me know in the comments.

Katie said...

I don't play the lottery. It would be great to win, but I see my mom buying tickets all the time and I always wonder how much money she has wasted on those things. Of course if you never play you never have any chance of winning but the odds are just too high for me.

I buy lottery tickets at Christmas as stocking stuffers and sometimes once or twice throughout the year but I never actually think I'll win. Especially that much money. The most I've won is \$8, lol!

Christa @ momvesting.com said...

That's a very good explanation of lottery odds -- one I could actually follow! I won't be buying a ticket. I never even win a buck...

The explanation is good, but it turns out in the case of huge lotteries the "nuances" are the whole deal. Since the nominal return on buying one ticket with every number combination is so huge (\$500M expected minus \$175M in ticket costs) there's a good chance that various professional gambling syndicates are trying to do exactly that. The difficulty of course lies in the logistics of purchasing 175M lottery tickets. But the very fact that people are trying likely in turn increases the chance of multiple winners. Which decreases the return.

So it's really a complicated game of chicken - if you're the only gambling syndicate that buys the whole ticket space, you're pretty much guaranteed the score of a lifetime. If you fail to cover the whole range of tickets, or it gets split 4-5 ways, you could lose 8-9 figures.

Tough game.

Bach said...

I will NOT be playing! I do buy scratchers for X-Mas stocking stuffers, and have bought one lottery ticket ever (two months ago I was in the gas station and felt this compelling feeling to buy one even though I am not a gambler- took it as a sign since it was my first instinct to "play" in nearly 30 years of life and... I did not win. :) a sucker born every minute I guess. If those folks who spend \$10 every week would just invest the money instead they'd see a much better return on their investment! Thanks for stopping by my site, glad I stopped by yours!

Annabelle said...

I've always felt very strongly that if I bought one lottery ticket, I would have to keep buying lottery tickets and so I never started. It's the same reason I'd never use a slot machine - I know once I started, I wouldn't be able to stop until I won something!

Bryan said...

Katie: I tend to agree. Playing the lotto is basically flushing your money away, and if you have someone close to you who does it, it's that much harder.

Daisy: When I turned 18, I spent \$20 on scratchers. I won, I think \$28, with which I bought more scratchers. I lost all that I'd won. If I stopped with that first \$8 win and never gambled again, I could have stayed ahead and been one of the few gamblers who made money.

Christa: It's definitely good to stay away from the lotto.

W: I like your comparison to a game of chicken. I wouldn't be too surprised to see multiple winners tonight. I've even heard that Greece was using bailout money to play the lottery.

Bach: It's definitely a better idea to just sock the money away than to spend it on the lottery.

Annabelle: It's a slipper slope. I don't always play the lottery, but I do tend to play when the prize gets above \$100 million. As such, I've been spending \$5 or so every drawing for the last few weeks, with nothing to show for it.