# Playing the Odds

**Posted:**March 28, 2012

**Filed under:**Math |

**Tags:**improbability, megamillionaire, not sports, numerology, yes i'm really writing about this 8 Comments

A colleague pointed out to me today that the NY MegaMillions jackpot is now up to $500 million. Of course after taxes, the most anyone could take home is a mere $375,000,002 (or $269,250,000 if the winner takes the lump sum option). And while the Forbes Top 400 no longer has anyone worth less than a billion on there, a half billion dollars is a totally absurd amount of money to make on a mostly unproductive investment.

The last time I can recall buying lottery tickets was when I was 7 or 8 visiting my grandparents in Kentucky, where lotteries were legal unlike my more ethical home state of Alabama. My dad and I would usually pick numbers together, but it occurs to me now that this was a ploy by my parents to get me out of the house so my grandmother could watch Wall $treet Week—the only show besides Jeopardy I remember her watching—without interruption.

As a game, the lottery is simple and uninteresting. They’re generally a horribly regressive form of taxation, but one that people willingly get in line to pay. In New York, while the lottery fund was historically designated to benefit public education, apparently what they do with the money is discretionary (or so a colleague tells me), as the New York Lottery has its own autonomous unit within the state Department of Taxation and Finance. Maybe the only interesting thing about them are the sociological facts about who decides to play or which states’ fundamentalists are opposed to what they consider an unethical form of gambling. The odds of winning are essentially zero, and no strategy of choosing numbers can affect that other than buying up giant stacks of tickets.

But as Felix Salmon pointed out a little over a week ago when the payout was a paltry $241 million, at some point the payout is so great that buying lottery tickets becomes an economically rational decision. Shocking given the odds, but in the New York lotto, that point has been passed.

An interesting thing happens, when the jackpot gets this big: if you actually believe the $241 million figure, the expected return on your dollar is

positive.The mean player, it turns out, is going to get paid out to the tune of $1.553 for every ticket they buy. In reality, sadly, the cash option is $170 million, which brings the expected payout to $1.149 per dollar spent, which means that after taxes, you still have to expect to lose money.

Now that the one-time cash payout is in excess of $241 million, we are in the realm where even with taxes, the return on every lottery dollar spent is positive. That’s in part because there are several prizes other than the jackpot. So until these odds flip back, I might be spending a few spare dollars buying MegaMillions tickets. Now I just have to decide whether to fill in my own numbers or go with the auto-generate.

Because I am a nerd, I did my own odds calculation for the NY lotto a couple years ago. It worked out so that whenever the jackpot was over $169 million, it was worth it to play. I still go by that figure–when I notice that the jackpot is over $169 million, I’ll invest (yes, invest) in a ticket.

What’s the thinking on the lump sum versus the 26 annuity payments?

I played around with some numbers, and as long as you’re getting 1.25% interest on the lump sum, you’d more than make up the difference over 26 years. Of course to really compare them, you’d also need to make up for the additional interest you’d be earning on the annual payments, but you’d still only need a return in the 2% range.

The other thing is that annuities aren’t adjusted for inflation either, which is currently in the 3% range. So the present value of the additional money you get with the 26 definitely comes out to be a lot less money if you can manage the lump sum even decently well. Am I missing something?

Let’s pool some money, kids.

I’m getting some tickets on the side, but we could do a pool of like $10/person. You want to buy some tickets tonight?

I’d join in a pool of tickets. I may buy some during my lunch break tomorrow–let me know if you guys have made any purchases by then.

A mathematician over at Slate when into more depth on the odds. Adjusting for the possibility of multiple winners, still a positive return on every dollar spent.

http://www.slate.com/articles/life/do_the_math/2001/08/is_powerball_a_mugs_game.single.html

Also, now the jackpot is at $540 million.

I wonder if/how the poker concept of implied odds could be used for the lottery. It’s not a perfect analogy, but despite the overwhelming odds against of ever winning the lottery, is it worth it to play because the pay off is worth it?

That’s exactly the thinking. Your odds of winning are still lower than the probability of getting hit by lightening. But multiplied by the size of the pot, the value is in excess of the $1 that a ticket costs.