Update on January 11, 2016: The interest in this page has become intense in the last few days as the current Powerball jackpot is being advertised at over $1.4 billion. I estimate there will be roughly one billion tickets in play trying to win the jackpot. Keep in mind that this number is speculative: it extrapolates from previous smaller jackpots. Who knows what will really happen?

Unfortunately for those of you hoping to buy a sports team, a billion tickets sold means the expected value of a $2 ticket is only about 71 cents (or a bit more or less, depending on your state).

On the plus side, the frenzy will almost certainly be over after Wednesday's drawing. There will be probably be about three times as many tickets sold as there are possible number combinations on a PowerBall ticket! There will almost certainly be a winner, most likely a tie between two or more winners. To be more precise, with a billion tickets in play:

Keep in mind the probability of winning is, as always, fantastically low. Imagine someone were to lay down a strip of pennies along the road from Seattle to Miami, and put a secret "X" on just one of them. You are about as likely to find the "X" by randomly picking a single one of those pennies as you are to win the Powerball jackpot.

Summary

In the past year, the Mega Millions and Powerball lottery jackpots have shattered records, rocketing past half a billion dollars each. Should you have bought a ticket? Would any size jackpot make a ticket an economically rational investment?

Mega Millions and Powerball are a popular multi-state lotteries in the United States that use what's called a "progressive jackpot." If no one wins the jackpot in a lottery drawing, the prize money is carried forward into the next drawing's prize pool, combined with new money from the previous drawing's sales. Most people know that lotteries are losing propositions under normal conditions, but progressive jackpots can, in theory, grow without bound. Since the probability of winning a jackpot remains fixed, an intriguing possibility emerges: might it actually become profitable, in expectation, to buy a lottery ticket?

Many people have already observed that a lottery ticket purchased in isolation will have a positive expected value if the post-tax jackpot grows larger than the odds of winning. However, larger jackpots generate higher ticket sales, and in the case of a tie the jackpot is split among all winners. Some analyses account for the effect of ties, but none that I've found have observed that as the jackpot grows, ticket sales appear to increase super-linearly. This means there is a point of diminishing return where the negative expectation due to ties outweighs the positive expectation due to having a larger jackpot. Taking this into account, it is unlikely that buying a lottery ticket is ever profitable in expectation, no matter how big the jackpot gets.

Winning If You're the Only Player

Let's start with the simplest case. Imagine that you are the only person buying a Mega Millions ticket. A ticket costs $1, and the probability of winning the jackpot is 1 in 175,711,536. Therefore, if the jackpot is expected to yield exactly $175,711,536, then your $1 ticket has an expected value of $1 -- it's an even-money proposition, albeit one with very high variance. To simplify the calculations, assume for now that the jackpot is the only prize; we'll consider the relatively small effect of the smaller prizes in a moment. I'm also assuming there are no intangible benefits to playing, such as the thrill you get from watching the draw. And keep in mind just how low the probability of winning is: imagine someone has laid down a strip of pennies along the 2,000 miles (3,200 km) of highway from Los Angeles to Chicago, and put a secret "X" on just one of them. Take a road trip, stop randomly on the side of the road, and pick up a single penny. You are about as likely to find the "X" as you are to win the Mega Millions jackpot.

But this is no time for such negative thinking! It might seem at first glance that any time the jackpot grows beyond $175.7 million, it's economically rational to buy a ticket. Buying a ticket for March's $640 million drawing would have been a smart move, right?

Slow down, big dreamer. Unfortunately, an advertised jackpot of $175.7 million doesn't actually yield $175.7 million in prize money. There are two important deductions to take into account:

So the world-record $640 million jackpot in March of 2012 wasn't actually worth $640 million after all. Taking tax and present value into account, it was really "only" worth $640 x 0.63 x 0.59 = $238 million if you'd been the only person playing. That's still a lot -- and it would make the expected value of a ticket $1.35. Seems like a smart bet so far!

Let's also consider the other, non-jackpot prizes. They are paid immediately (without the 26-year annuity), so we only need to discount them for taxes, not present value. Multiplying each prize by the probability of winning it yields its contribution to the expected value of a ticket:

Prize Post-Tax Prize Odds of
Winning		 Post-Tax
Expected Value
$250,000 $147,500. 1:3,904,701 $0.0378
$10,000 $5,900. 1:689,065 $0.0086
$150 $88.5 1:15,313 $0.0058
$150 $88.5 1:13,781 $0.0064
$7 $4.13 1:306 $0.0135
$10 $5.9 1:844 $0.007
$3 $1.77 1:141 $0.0126
$2 $1.18 1:75 $0.0157
Total $0.107

The non-jackpot prizes add a total of 10.7 cents of expected value to a ticket, independent of the jackpot size. Added to the $1.35 of expected value from the jackpot, a ticket for a $640 million jackpot purchased in isolation would be worth an expected $1.46 -- more than its cost of $1! Were those people standing in line for two hours to buy tickets doing so for good reason?

Wait a second. I keep saying a ticket would be worth $1.46 when purchased "in isolation," but all those people in line were buying tickets, too. What if you're not the only person playing?

The problem of ties

The analysis so far assumes you're the only one buying a ticket. The problem is that you aren't. If more than one ticket hits the jackpot, the prize is split equally among all the winners, reducing its value to each winner.

To make the problem concrete, let's look at the $640 million jackpot in 2012 again. According to LottoReport.com, a site that has recorded the sales figures for every Mega Millions drawing since 2007, there were nearly 652 million tickets sold to hopeful future private jet owners. Let's call this n. Each ticket was independent and had a 1 in 175.7 million chance of winning. Let's call this p. The Poisson distribution with λ = np = 3.71 tells us how many winners we could expect in that drawing; it's shown in the plot on the right (click to enlarge).

There was a slim 9% chance that a single winner would keep the entire jackpot, an 88.5% chance that there would be a tie among two or more winners, and a 2.4% chance that the jackpot would remain unclaimed and grow still larger. The two most likely outcomes were a three-way tie (20.8%) and a four-way tie (19.3%). (The actual result: a three-way tie.)

The overwhelming likelihood of a tie had a devastating effect on the expected value of a ticket. The most straightforward way to compute the jackpot's contribution to each ticket's expected value, suggested by Mark Eichenlaub, is to divide the expected total payout by the number of tickets in play. The expected payout is just the jackpot size multiplied by the probability that at least one ticket would win -- 0.975, in this case. Accounting for those pesky taxes and net present value, the jackpot only contributed a paltry 35.6 cents to the value of a ticket in that drawing!

Using some fancier math, we can dig a little further into why having so many tickets in play pushes down the expected value of each one. Let's assume you'd been holding a winning ticket. There was only a 2.4% chance that not a single one of the other 652 million tickets out there won, too. We can repeat this analysis for every realistic number of winners: compute the probability of that kind of tie occurring, calculate the value of the jackpot per winner in that case, and multiply those two numbers together to get that scenario's contribution to the expected jackpot. Add them all up, and we have the expected value of the jackpot, taking ties into account, as shown in the table below:

Winners
(Including You)		 Probability Jackpot Share
($millions)		 Jackpot Share
Post-Tax NPV		 Contribution to
Expected Jackpot
($millions)
1 0.0244739 640. 237.89 5.82
2 0.0908017 320. 118.94 10.8
3 0.168444 213.33 79.3 13.36
4 0.208317 160. 59.47 12.39
5 0.193222 128. 47.58 9.19
6 0.143377 106.67 39.65 5.68
7 0.0886581 91.43 33.98 3.01
8 0.0469907 80. 29.74 1.4
9 0.0217928 71.11 26.43 0.58
10 0.00898383 64. 23.79 0.21
11 0.00333314 58.18 21.63 0.07
12 0.00112422 53.33 19.82 0.02
Total 62.55

With 652 million tickets in play, a 1 in 175.7 million chance of each ticket winning, a 63% net-present value, a 41% tax burden, and an equal split among multiple winners, the expected value of a $640 million jackpot to a winner is just $62.55 million, making the jackpot's contribution to the expected value of a ticket only 35.6 cents. Adding back the 10.7 cents of expected value from the non-jackpot prizes, the total expected value of a ticket in that drawing drawing was just 46.3 cents! This is far worse than the $1.46 we computed when we ignored the possibility of a tie. The expected value of a ticket was actually far below its $1 cost. It wasn't such a smart buy after all.

Ticket sales as a function of jackpot size

Unfortunately, we are still not done dumping cold water on the dreams of lottery winners. (It's a cruel hobby, I know.) The analysis above took ties into account, but assumed a fixed number of tickets in play -- 652 million, to be exact. However, as the media delights in reporting, large jackpots incite a ticket buying frenzy, and that makes ties increasingly likely. Might the increased possibility of a tie actually make larger jackpots less valuable to the winners?

The first step in answering that question is to model how the size of the jackpot will end up affecting the number of tickets that are bought to try to win it. Let's return to the excellent data curated by LottoReport.com. The red points in the graph on the right (click to enlarge) are ticket sales for 618 MegaMillions drawings since 2007, plotted against the drawing's advertised jackpot. The lonely point way out at the top right is the record-breaking March 2012 jackpot of $640 million, selling 652 million tickets. The blue line is the best-fit third degree polynomial.

As you can see, with jackpots over about $250 million, ticket sales really start to take off. However, it's also important to acknowledge that the data for jackpots above $350 million is sparse, and non-existent above $640 million. The curve may not be predictive for unprecedented jackpots. If we ever see a mind-boggling billion-dollar jackpot, for example, it predicts frenzied buyers will pay for a staggering two billion tickets. But there's no way to know with certainty what would actually happen if a MegaMillions jackpot grew to such proportions. Mass hysteria?

The projected sales curve above also paints a grim picture for anyone still holding out hope that a lottery ticket can ever be an economically rational investment. As the jackpot grows in value, the number of people who try to win it grows super-linearly. This human behavior has a mathematical consequence: 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. The graph below shows the effect, plotting advertised jackpot size against expected jackpot value if you win. It takes everything into account we've discussed so far: net present value (63% of jackpot), taxes (41% off), and the increasingly devastating effect of ties as the number of tickets grows super-linearly with jackpot size.

The right-hand axis in the above plot shows the expected value of a ticket at each advertised jackpot size; it is the expected jackpot value times the 1/175.7 million probability of winning the jackpot, plus the 10.7 cents of expected value that comes from the non-jackpot prizes. It peaks at 57.8 cents when the advertised jackpot reaches $385 million. The $640 million jackpot in March was already on the downward part of the curve; a ticket for that drawing was actually worth less, in expectation, than a ticket for the $363 million jackpot that preceded it! As long as the superlinear ticket growth holds up, tickets for even bigger jackpots will be worth even less. In no case does the expected value of a ticket ever exceed the break-even point of $1. Thus, Mega Millions tickets are never an economically rational investment, no matter how big the jackpot grows.

Powerball

Powerball is similar to Mega Millions: a multi-state lottery with a progressive jackpot that is split equally among winning tickets. For most of its history, tickets cost $1. However, they've made various rules changes since 2012, including raising the price of a ticket to $2 and making tickets more likely to win small prizes and less likely to win the jackpot. (The Powerball jackpot odds are now one in 292 million, much lower than its former value, nearly identical to MegaMillions, of 1 in 175 million.) This was clever: selling half as many tickets at twice the price might seem to be revenue-neutral, but fewer tickets and lower odds of hitting the jackpot means fewer jackpot winners. Fewer jackpot winners makes larger jackpots more likely, and large jackpots produce huge spikes in ticket sales. This effect is important to our analysis because fewer tickets in play also means fewer ties and more money, in expectation, for a winner. Might Powerball hold the key to financial salvation?

Unfortunately for those of you hoping to buy a sports team, Powerball turns out to be a bad bet, too. As with Mega Millions, Powerball's ticket sales (again, from LottoReport) are superlinear, shown in the graph on the right (click to enlarge). Included are sales data between the ticket price increase (January 2012) and now (January 2016). The point way out on the top right is the record-breaking $948 million jackpot on January 9, 2016, that sold 440 million tickets. That point has been given additional weight in the curve to help better project the behavior with large jackpots.

Due largely to the $2 tickets producing fewer ticket sales, the effect of ties is delayed compared to Mega Millions. For Powerball, the best ticket value is for jackpots advertised at about $890 million. We'd expect to see about 390 million tickets in play, for which there'd be a 26% chance of no winner, a 35% chance of a single winner, and a 39% chance of a tie among two or more winners. The expected value of the jackpot to a winner, accounting for ties, and discounting for taxes and net present value, would be $180 million. Adding in the expected value of non-jackpot prizes of 19.1 cents, this gives the ticket an overall expected value of 80.8 cents -- less than half of its $2 cost. A good deal for state governments' ailing tax coffers, perhaps, but certainly not one for players.

Can the models really predict the future?

It's easy to predict how many tickets will be sold for a drawing when the jackpot is less than a few hundred million dollars. We have plenty of data for those drawings showing consistent sales, so interpolation works well. But how confident can we be in the model when it extrapolates to jackpots larger than our past experience?

We've had two opportunities to evaluate the sales model's extrapolation performance since I first wrote this article in January of 2011. At the time, the record for the largest MegaMillions jackpot was $380 million. The following year, in March of 2012, the jackpot grew to an unprecedented $640 million. Ticket sales skyrocketed due in part to intense media coverage, some of which even described my analysis of why tickets were still expected to be unprofitable: The Wall Street Journal's blog and print editions, the Washington Post, the Atlantic, and Time Magazine.

The 2011 model predicted a Mega Millions jackpot that size would produce $748m in ticket sales. According to LottoReport, actual sales were $652m, an error of only about 15%. Based on the drawing's actual ticket sales, the expected value of a ticket was 46 cents; the model predicted 42 cents. (10.7 cents of EV came from non-jackpot prizes in both cases.)

Another test came in November of 2012 when the Powerball jackpot grew to a record $587 million. I had never studied Powerball before, but quickly put together an analysis after NBC's Today Show asked me to comment in the segment they aired the morning of the big drawing.

Powerball had changed their rules less than a year earlier, raising the ticket price to $2 from $1 and adjusting the prizes to match. This seemed likely to impact sales, so I only used data from after the rule change in my model. Unfortunately, this left me with only 90 data points, and this model did not perform as well as the MegaMillions model. 281 million $2 tickets were sold to win the big jackpot, but the model predicted 54% more: 435 million. I'd predicted a ticket's expected value for that big drawing would be 67.3 cents; in reality, it was 83.3 cents -- still well short of its $2 price.

The models weren't perfect, but I think their fundamentals are sound, and they're much improved from 2011 now that we have new data points for huge jackpots. But, perhaps more importantly, these two tests demonstrated the most important feature of my analysis. Before the big 2012 jackpots, every lottery we'd seen had followed the rule that a bigger jackpot made for a more valuable ticket. The idea that superlinear sales would force the value of a ticket back down again was speculation based on my model of sales trends. But we've now seen two jackpots above half a billion dollars, and my prediction came true: sales were so strong that the negative effect of ties overwhelmed the positive effect of having a larger prize. These two drawings are the first that are past the peaks seen on the jackpot-vs.-value graphs -- proving that the peaks exist.

Still, it bears repeating that this analysis rests on our ability to predict future lottery sales based on the history of previous sales. Past $640 million, we have no evidence. Who knows what will really happen when a billion-dollar jackpot finally happens? Maybe with a big enough prize, the market will saturate -- everyone who knows how to buy lottery tickets has bought all the tickets they can afford and the sales curve gets flat.

For the last word on this topic, however, I cede the floor to Durango Bill, who aptly observes that driving to the store to buy a Mega Millions ticket is more likely to be fatal than it is to make you rich.

That's why I plan to walk.

Jeremy Elson is a computer science researcher who spends his time outside of work riding bicycles, flying airplanes, building electronics, and playing with Mathematica. (This work is a personal project; I do not speak for my employer.)