Hacking The Powerball Lottery

This item has been corrected.

The jackpot in the Nov. 28 Powerball lottery, a drawing across two-thirds of the United States, is up to a record $500 million. And while it’s never a good idea to buy a lottery ticket, if you insist, now may actually be the best time to go all-in on this investment. Let’s break down the odds.

Lottery players have long dreamed of making the perfect bet: buy every possible combination of numbers to guarantee yourself a jackpot and, if the pot is large enough, profit the difference. Powerball tickets cost $2 each, and you can play the numbers 175,223,510 different ways, so with a sizable but short-term loan, you ought to pocket about $150 million on this investment, right?

Well, no. There are taxes to consider, and prizes aren’t paid all at once, but more to the point: as jackpots grow and the number of tickets sold increases, so do the chances of multiple winners splitting the pot. In short, it’s not a good bet.

Here’s a chart by Jeremy Elson showing a strong correlation between the jackpot size and ticket sales for a very similar American lottery, MegaMillions:

So unless you can keep the half-billion-dollar Powerball jackpot a secretâ€"good luck with thatâ€"ticket sales are going to be undermining the scheme, rendering it more likely we’ll see multiple winners than one or none. As Elson put it, “Even though the jackpot itself can theoretically grow without bound, there is a point at which the consequent ticket-buying grows to such a fever pitch that the expected value of the jackpot actually starts going down again.“

It’s a bell curve. Here, again, is an Elson chart for MegaMillions, which has nearly identical odds but tickets that cost only $1. The analysis accounts for taxes and the haircut taken on a one-time cash payout. (For the current Powerball drawing, the $500 million jackpot is worth just $327.4 million if you take the prize all at once.)

The peak of that curve is a $420 million jackpot, when the “expected value” of a $1 ticket is 69.3 cents. Not great, but better than investing in the Groupon IPO.

Now, the values in the chart above don’t apply to Powerball, which costs $2 per ticket, but it would still be a bell curve. And because the higher price of Powerball likely leads to fewer tickets purchased, the peak of the curve will be further to the right. Which means that the expected value of Powerball tickets for this week’s record jackpot has probably never been greater.

So not that we’re recommending it, but now is as good a time as any to play every possible combination and pray for the best. The odds aren’t in your favorâ€"in fact, even if you’re the only winner, you probably won’t break even after taxesâ€"but it’s the best shot you’ve ever had.

Correction (Nov. 27, 8:40 p.m. ET): An earlier version of this article said that a Powerball ticket costs $1, but the price was raised to $2 from $1 at the beginning of 2012. This article has been corrected in several places to reflect the accurate ticket cost and the ways that affects your odds.