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### The Powerball Jackpot is \$425m. Should you play?

#### Everything you never needed to know about Powerball

Once again, a large lottery jackpot is making headlines around the country. Tonight’s Powerball jackpot is currently advertised at \$425 million. You might be wondering to yourself, “hmm, I wonder if the jackpot is big enough to be worth playing?” Before you go out and buy a ticket, why not take a look at the math?

### The math in all the headlines

Tonight at 10:59 p.m. EDT, at Universal Studios in Orlando, Florida, a machine will select the winning Powerball number by selecting 5 white balls out of a drum of 59 and one red ball out of a drum of 35. That means the total number of possible combinations is 59 choose 5 for the white balls times 35 choose 1 for the red balls, for a total of 175,233,510. It costs two dollars to play. So if the Jackpot is more than \$350,467,020, it makes sense to play, right?

Well, not exactly.

First off, that \$425m in all the headlines is an annuity that gets paid out over 30 years. That \$2 is coming out of your pocket is actual money, so you want to compare it to the actual money you win. The Powerball people advertise this number, too, but in smaller print: \$244.7m. Not a bad return on a \$2 ticket, but it’s a lot less than the \$350m we set as our “worth playing” threshold. So you shouldn’t play until the jackpot gets bigger, right?

Again, not exactly.

### Small prizes

Even if you don’t win the jackpot, you can win smaller prizes. Your ticket needs to match all six balls to win the jackpot, but if you match the red ball and/or three out of five red balls, you win something. Outside of California (which we’ll get to later), there are 8 lower prize tiers ranging from \$4 (for matching just the red ball or three out of five white balls) to \$1,000,000 (for matching all five white balls but not the red ball).

The total expected value of those smaller prizes is about 36 cents per ticket. So to calculate the break-even jackpot point, you could say that on a \$2 ticket, you’re spending \$0.36 for small prizes and \$1.64 on a shot at the jackpot. So the break-even jackpot would be 175,223,510 x \$1.64 = \$287,366,556 (actually, since it’s not exactly 36 cents worth of small prizes per ticket, the number is \$287,280,900). We’re not there quite yet, but we’re close, right? If no one wins tonight, the jackpot will go up past the \$287m threshold. Does it make sense to buy then?

As you’ve probably guessed, it gets more complicated.

### Split Jackpots

If you do win the jackpot, there’s a good chance you won’t take home the whole prize: there are a lot of other people buying tickets, and there’s a good chance that one or more of them will split the jackpot with you. Exactly how good a chance depends on how many people play. So, how many people play?

Well, Saturday night’s advertised (annuity) jackpot was \$300m. Discounting to the cash value, that’s \$172.7m. Meaning that the cash value of the jackpot has grown by \$72m since the last drawing. Each ticket purchased nationwide adds \$0.5995 to the jackpot, so there must have been 120 million tickets sold for tonight’s drawing. If the tickets are randomly distributed among the 175 million combinations, then there’s a (1-1/175,223,510)^(120,000,000) = 50.4% chance that you get the entire jackpot to yourself.

If there are C total possible ticket combinations and N total tickets sold, then if tickets are randomly distributed, the probability of splitting the jackpot with S other people is the binomial function: ((1/C)^N)*((1-1/C)^(N-S))*(N choose S). Because we’re dealing with such large values of N and C, this can be approximated as P(S)=((1-1/C)^N)*((1-1/C)^(-S))*(C^(-S))*((N^S)/S!) = ((1-1/C)^N)*((N/C)^S)/S! In this case, C= 175,223,510 and N=120,000,000, so the formula works out to (50.4%)*0.685^S/S!

If you split the jackpot with S other people, then your payout will be divided by (1+S), so to calculate the expected value of the jackpot if you win, you need to sum, from S=0 to (in theory) S=N, P(S)/(S+1). In reality, P(S)/(S+1) gets vanishingly small rather quickly, and it’s easy to calculate that the sum is 0.724, meaning that if you win the jackpot, you should only expect to win 0.724*\$244.7m = \$177.2m. So the expected value of a \$2 ticket is \$0.36 for the small prizes plus \$1.01 for the large prizes, for a total of \$1.37.

From here, it’s tempting to just say, “well, sure, we just need the jackpot to be high enough so 0.724 times the jackpot is more than \$287m. \$397m, or a headline (annuity) jackpot of \$689 million, ought to be the break even point.

But there’s a problem with that logic. The higher the jackpot grows, the more people buy tickets. And the more people buy tickets (bigger N), the lower the sum of [((1-1/C)^N)*((N/C)^S)/(S+1)!] gets. Since Powerball expanded to California, there has only been one headline jackpot larger than the current \$425m: a \$590m jackpot that sold roughly 240 million tickets. At N=240 million, the split jackpot correction factor drops to 0.545, and at that correction factor, the jackpot would have to be \$287m/0.545 = \$527m cash, or a headline jackpot of \$915m. Taking the probability of a split jackpot into account, the expected value of a \$2 ticket on that lottery was still only \$1.52.

That is far higher than any jackpot on record (excluding the “El Gordo” prize in Spain’s Christmas lottery, but that’s generally divided into 180 smaller shares), so we don’t have any data on how many people would buy tickets. It certainly seems, though, that the trend of higher jackpots, leading to more sales, leading to a higher likelihood of a split jackpot, would continue.

Which leads to the surprising conclusion: It’s not clear that there is any possible Powerball jackpot size where buying a ticket has positive expected value.

### Taxes and Utility

It gets worse.

In the United States, money you win in the lottery is taxable income. And for most people, money you spend on lottery tickets isn’t tax deductible. So you’re wagering post-tax dollars to win pre-tax dollars.

Now, a disclaimer: I’m not a tax expert, so don’t take this as tax advice. But in general, the federal government and many states allow you to deduct gambling losses up to the amount of your gambling winnings for the year. So if you already have a net gambling profit for the year, you can probably deduct the cost of your \$2 Powerball ticket. There’s still probably some mucking about to do about possible differences in marginal tax rates, AMT exclusions, and such, but like I said, I’m not a tax expert. To a first-order approximation, even if the expected value of a ticket does exceed \$2, unless you’ve already got gambling winnings this year, after taking taxes into account, it’s still a losing proposition.

And even aside from taxes, there’s the matter of the utility a person gets from money. \$200,000,000 is not actually a hundred million times as useful as \$2. Without getting too deep into the philosophical questions about money and happiness, let’s just go with a simple model where the utility of money is the log of total wealth. If we’re trying to optimize expected utility, then if your current total wealth is W and the probability of winning the jackpot is P, you should play when ln(W) < ((1-P)*(ln(W-\$1.64)) + P*(ln(W+J))). This is approximately the same as J > (W/(W-\$1.64))^(1/P)-W.

1/P is just the number of possible ticket combinations, 175,223,510. If the median American has \$70,000 in wealth, then they can maximize their utility by playing when the jackpot exceeds ((70,000/69,998.36))^(175,223,510). This is a number with well over a thousand digits. Even if you’re a millionaire, the jackpot necessary to justify playing is well over a hundred digits. You need to start with a total wealth of more than \$10 million to ever possibly justify spending \$2 on a Powerball ticket, if you’re trying to maximize your expected utility of wealth.

So that means you should never buy a Powerball ticket unless you’re rich, already have gambling winnings for the year, and the expected jackpot is really high, even after taking the probability of splitting the jackpot into account. Right?

Well, again, not quite.

### Other considerations

Here’s where I come clean: in spite of all the math, I’ve actually bought a ticket for tonight’s drawing.

Why?

Because Powerball’s not an investment. It’s a game. I play because I think it’s entertaining. It’s like playing whack-a-mole at the carnival: maybe you’ll win some cheap trinket, but odds are the only thing you’re paying for is the experience of playing.

As you’ve probably figured out by this point, I enjoy thinking about the math related to the lottery. I also enjoy letting my mind wander and think about what I’d do with a nine-figure windfall. And I find it a little easier to let my mind ponder these questions when I know I’ve got a lottery ticket in my pocket. These aren’t things I need to think about every day, so I don’t play the lottery every day, but spending \$2 a couple of times a year is worth it for me.

Also, of that \$2 ticket, about \$0.96 goes into prizes, about 10 cents goes to the local business where I bought the ticket, about 5-10 cents goes to lottery administration expenses, and the rest goes to the state. And I know I’m in the minority here, but I really don’t mind giving the state government a few extra dollars each year. I think most of what the government does is pretty worthwhile.

At the end of the day, \$2 just isn’t that much. You probably wouldn’t think twice about spending that much on bus fare, or on a bottle of water. If you feel like buying a ticket, buy a ticket, if you don’t, don’t, but remember that it’s entertainment, and not investment.

### California is special

As anyone who lives here knows, California is a little different from the rest of the country. As it turns out, California’s Powerball rules are also a little different from the rest of the country.

In most states, the smaller Powerball prizes are fixed amounts specified in the rules. For example, if you match only the red ball, and no white balls, you win \$4. The odds of winning that prize are 1 in 55.406, so the expected value of that prize is 7.2194 cents. Due to random variation, sometimes more people win prizes, and the lottery pays out a little more than 7.2194 cents per ticket sold, and sometimes fewer people win, and they lottery pays out a little less than 7.2194 cents per ticket sold.

In California, however, prizes are calculated in the opposite direction. The lottery allocates exactly 7.2194 cents per ticket sold to the “Level Nine” prize pool, and then divides that prize pool evenly among everyone who matches just the red ball and no white balls. When fewer than the expected number of people win prizes, then the quotient is a little higher than \$4, and when more than the expected number of people win prizes, the quotient is a little lower than \$4. But the California Lottery always rounds down to the next whole dollar, with the remainder going into a “special prize fund” used to promote other games. So about half the time you win a small prize, you’ll get a dollar less than you expected. On net, this decreases the expected value of California Powerball tickets by 1.57 cents relative to non-California Powerball tickets.

There’s another quirk in the California rules, too: while all the other Powerball states award a \$1,000,000 prize for matching all five white balls, but not the red ball, California treats that prize almost like a separate jackpot game. Other states’ expected payout on the million dollar prize is 19.4037 cents per ticket, but sometimes it’s more, and sometimes it’s less. California, on the other hand, allocates exactly 19.4037 cents per ticket sold to the “Level Two” prize pool. And fairly often, nobody in California wins this prize. When that happens, the prize pool carries over to the next drawing. The exact adjustment is a function of how many tickets were sold for the drawing statewide and how big the prize pool carryover is from the previous drawing. From past drawings, it appears that a million dollars in growth in the national headline jackpot corresponds to about 115,000 tickets sold in California, so for tonight’s drawing, there are probably about 14.4 million tickets in the state. And someone matched 5 out of 5 white balls on Saturday, so there’s no carryover from last time. The odds of matching 5/5 white balls and no red ball are 1 in 5,153,633. So there’s an additional downward adjustment of \$0.194037*((1-1/5,153,633)^14,400,000) = 1.19 cents per ticket. So, overall, California Powerball tickets for tonight’s drawing are worth 2.76 cents less than non-California Powerball tickets.

### TL;DR

The one sentence takeaway: no matter how big the jackpot gets, you should only buy Powerball tickets for their entertainment value.

And if you buy a ticket, good luck tonight.