Sorry, you cannot guarantee a Powerball win

Forget MegaMillions — how about a Power Billion?

The jackpot for Wednesday night's drawing of the Powerball lottery is now estimated at $1.5 billion, the highest U.S. payoff ever. With a potential windfall like that, it's easy to rationalize throwing down $2 for a ticket and a shot at a life of leisure.

In fact, it's theoretically easy to guarantee a profit. At least if you don't think about it too hard.

To guarantee a win, you'd need to buy a ticket with every single combination of numbers. That way, no matter what numbers are drawn, you'd have the winning ticket.

Powerball works by picking random balls to create a set of numbers. Five balls are drawn from a bin of 69 white balls, and one red ball (the "powerball") is drawn from a different bin of 26 balls. To win the jackpot, you have to pick numbers that match all five of the white balls, plus the red ball. (If you pick the five white balls but miss the powerball, you win a measly — yawn — $1 million.)

Doing the math, that means there are 292,201,338 different combinations of numbers you could pick.

At $2 a pop, you'd need almost $590 million to buy every ticket combination, which would bring you a tidy little profit approaching $1 billion in the current scenario. But finding the money would't be the hard part — the logistics of buying 292 million lottery tickets would.

With a lottery ticket machine running constantly, it would take nearly a decade to print all of those tickets, assuming a print rate of one ticket every second. There's just a few dozen hours before the drawing (and some states cut off ticket purchases hours before). Even if you had 10 machines going at the same time, it would still take a year of printing.

Even so, one second per ticket is way too short a time estimate. Leaving out the fact that a lottery machine printing tickets all day would run out of paper and ink — and that the mechanics would probably break down at some point — that one second per ticket would depend on the machines random-number generator. If you trusted in the random number generator to select your numbers (clearly quicker than filling out each sheet by hand) you'd surely get duplicate tickets.

In fact, the probability of drawing the 292 million different combinations sequentially at random is something like 1 in a gajillion (technical term here). Seriously though, it's basically impossible you can randomly pick that many numbers without a repeat. After just 20,000 tickets, your odds of a repeat are already 50/50. Forget about it beyond 1 million tickets.

Normally, it's a bad financial decision to play the lottery. "It's a tax on people who are bad at math," is a common refrain. That's because the payout you can anticipate from buying a ticket (the "expected value") isn't worth the investment. For example, if the jackpot is valued at $100 million, the expected value of a ticket is just 66 cents, far less than the $2 you'd be shelling out.

This week, with a $1.5 billion jackpot on the line, the expected value is increased to more than $5 (including the nonjackpot prizes), making a $2 ticket a good investment. Even the estimated cash value of the jackpot — around $930 million — generates an expected value north of $3.50.

There hasn't been a Powerball winner since Nov. 4, which is why the prize money is so big. Powerball has what's known in casinos as a "progressive jackpot," which means the money increases incrementally each time it's played.

But wait a second. The biggest U.S. jackpot in history is bound to attract interest from more players than just you. If you were the only person buying Powerball tickets with a $1.5 billion jackpot, you'd be a fool not to invest $600 million. But everyone and their mother is buying tickets for Wednesday's drawing, so the set of unique number combinations is reduced dramatically.

Up to 70 percent of the possible number combinations had reportedly been purchased for Saturday's drawing (jackpot: $948 million) and current estimates anticipate up to 80 percent will be bought for Wednesday's drawing.

So the chances of someone winning the jackpot is pretty good, as is the likelihood of at least two tickets sharing the big prize. After all, people tend to pick the same, predictable numbers for the lottery. And the number of people playing the lottery tends to increase with the jackpot size.

If multiple tickets share the winning combination of numbers, the players split the jackpot. So the expected value of a ticket drops accordingly. In fact, even with a $1.5 billion jackpot (ignoring for a moment the lesser cash value) the return on investment on your $600 million only makes sense if you share the prize with one other person. If three people split the jackpot, you're already in negative ROI territory.

So while the prospect of a billion-dollar jackpot might be appealing, you're far from guaranteed a profitable payday.