The Fascinating Math Behind Why You Won’t Win Powerball
According to recent research, the winning Powerball numbers are 86 75 30 9… Just kidding, that’s Jenny’s number. Not to say that would be the worst choice. Your odds of winning this week’s record Powerball jackpot with Jenny’s help, or any other combination of numbers, are pretty bad.
How bad? Like one in more than 292 million bad. Or put another way, barely better odds than having your name randomly pulled from a hat filled with the names of everyone in the US. But you’re an optimist, right? I say that and you figure I’m telling you there’s a chance. And behind every game of chance is some fascinating math.
Powerball is like normal lottery with a twist. You pick five numbers—ranging from one through 69—which pop up as white balls in the eventual drawing. A final, yellow Powerball only ranges from one to 26, but always comes last.
The one-in-292-million odds come from all the different combinations you can win by picking the right five white balls—in any order—out of a drum of 69. “Multiplied by 26, because for every five-ball combination you’ve got 26 different associated Powerballs,” says University of Buffalo statistician Jeffrey Miecznikowski. And for the jackpot, those odds never change. Your odds of winning tomorrow’s $1.4 billion (and counting) jackpot are exactly the same as they were of winning in November, when the pot was a mere $40 million.
Which means—at the very least—you’re better off buying a ticket now than you were then. This gets into something statisticians call “expected return.” Simply put, this is your odds of winning multiplied by the potential payout. If the odds were perfect, and you could never lose, then the expected return would be $40 million: a fantastic investment. If you buy one Powerball ticket, giving you a one in 292 million chance of winning $1.4 billion, your expected return is almost $5 on that $2 ticket—a great investment. But wait: Powerball’s expected return is actually way, way lower than that.
How low? First off, you only get $1.4 billion if you accept the payout over 30 years. If you want it all at once, your bounty drops to $868 million. Then you gotta pay taxes. I’m going to assume you’re not already rich, and therefore do not have a cabal of Randian lawyers standing between your money and the government. Which means you’re losing about 40 percent at the federal tax rate. Oh and also, the IRS withholds an extra 25 to 28 percent because you struck it rich gambling.
If you’re lucky enough to live somewhere without state taxes, you take home around $524 million. Otherwise, your pot could be as “little” as $394 million—but multiplied by 1/292,000,000, your expected return on a $2 ticket is about $1.79 at the high end, and $1.35 at the low. That’s all back-of-the-napkin math, but the answer tells you what you need to know: If your expected return is less than your investment, put your money towards something else.
Now, that expected return will get better as long as the jackpot keeps growing. Unless game goes on too long. “The higher the jackpot, the more people you can expect to play, so that $1.4 billion jackpot is more likely to get split in multiple ways,” says Miecznikowski.
Exactly how much those odds change is difficult to figure out, because you’d have to know exactly how many Powerball tickets have been sold. For obvious reasons, the Multi-State Lottery Association keeps that number under wraps. You could estimate based on the jackpot size. This week’s $1.4 billion is $450 million higher than Saturday’s prize. The jackpot is about 32.5 percent of the base buy-in rate, plus whatever was leftover from last time. At face value, that would mean more than 146 million people have bought tickets. However, that doesn’t account for people paying extra money for multipliers should they win.
And of course, these calculations don’t take human fallibility into account. “Not all combinations are equally likely to be chosen by people,” says Miecznikowski. For instance, people who choose their own numbers often use birth dates. This is fine when you are choosing the yellow Powerball, which only goes up to 26, but white balls go up to 69. That leaves 32 through 69 underrepresented. “And on Saturday, the Powerball number was 13. Superstitious folks are less likely to choose that,” says Miecznikowski. Your best bet, he adds, is to let the computer choose your numbers.
By the way, these long odds are by design. In October 2015, the non-profit organization controlling Powerball changed the format, increasing the range of white balls from 59 to 69, while lowering the range of Powerballs from 35 to 26. The odds dropped from one in 175 million to one in 292 million. This simultaneously did two things: made it easier to win low-stakes payouts (guessing just the Powerball nets you $4), and much, much harder to win the jackpot. This incentivizes more people to play, and makes the game harder to win. “By making the odds more difficult, it guaranteed record jackpots because it is rolling over every time,” says Miecznikowski. “The publicity that Powerball has gotten setting record jackpots has really fueled popularity of the game.”
In other words, you’re playing against a bigger and bigger crowd. But hey, may the odds be ever in your favor.