## Is a Billion \$ Powerball Finally Profitable to Play?

TL;DR Nope.

Highlights
• With the latest odds and given the way ticket purchases grow with the expected jackpot, the expected value of Powerball is negative. Even more so with a billion dollar payout.
• There is a 95% chance that there will be 3 or more winning tickets after the next draw. Almost zero chance no one will win.
• Don't laugh at the premise! Powerball did have a positive expected value when the expected jackpot fell within a range of \$400m-\$650m. The game was changed to reduce the odds in October, 2015 which "fixed" that problem.
First of all big shout out to Walter Hickey at Business Insider for the pointer to the Powerball data (here:http://www.lottoreport.com/powerballsales.htm) in this post a few years ago (here: http://www.businessinsider.com/heres-when-math-says-you-should-start-to-care-about-powerball-2013-9).

This chart plots the relationship between expected value of purchasing a ticket to estimated size of a jackpot. The model used takes into account the dramatic increase in tickets purchased as the jackpot size increases.

My analysis shows that because of the exponential increase in the number of tickets being played and the likewise dramatic increase in the likelihood of sharing the winnings, there is never a point where one will break even on buying a Powerball ticket... Now that they've made it harder to win.

Previous to 10/07/2015 this is what the expected value looked like:

Notice anything funny? Yeah it used to have a range of values where the expected value was positive! That means if the expected jackpot was somewhere between \$400m-\$700m it was actually a real investment for you to play the lotto. When the Powerball odds were reduced though, this stopped being a problem for the lottery. (You can read more about it here: http://news.lotteryhub.com/powerball-odds-set-change-october-2015/). Depending on how someone plays, the expected value may not mean much. Specifically for people who play only a few times in their lives since they won't play enough for the Law of Averages to even out the times when a player lost. A lot. Even so, I use it here because it's a pretty simple and intuitive framework to use to understand the value of an investment.

Let's dig deeper into the game with the current (and harder to win) odds to understand why, even in the face of a billion dollar payout, Powerball is a net negative play with these new odds.

## The Data

I'm using data that goes back to just after California joined Powerball so that I have a pretty consistent population of Powerball players. From here I related the number of tickets sold to the jackpot's estimated value. I use the first value in lotto report's jackpot data since whether or not I play can't take into account the final jackpot amount before I know what it is.

Let's start by looking at how many tickets are sold as a function of the estimated jackpot.

Definitely not a linear fit which is probably what we would expect. After all, the more money on the line, the more people that are going to think it's worth playing.

## Sharing Winnings is for Losers

It might make intuitive sense that if the jackpot is in the billions, it has to start to become profitable. The problem is that as the jackpot gets so large, the number of tickets purchased grows so much that we become much more likely to have to split the jackpot. This chart shows the probability of splitting a jackpot i different ways depending on the jackpot amount.

As you can see, having only one winning ticket dominates until \$400m or so. Once we get to about a \$700m jackpot it's more likely we will split our winnings than not. This chart is generated using maths from the "The Math" section below.

## The Model

Using this quick Excel, model we can predict the expected value of the current Powerball jackpot (as I write this it is \$1.1B). Plugging 1100 into the model (x was divided by 1,000,000 already) we get an expected value of -\$1.24.

Since the Powerball odds were increased the highest expected value has been -\$0.44 at an expected jackpot of \$675m during the last draw on January 9th.

## The Math

Let's dig deeper and establish what we mean by expected value. We're going to get a bit mathy for those that would care to step through my thought process completely. Feel free to skip this part if it gets a bit boring though.

Let J be the total jackpot amount, T is the number of tickets which won, w_i is the event where a single ticket wins and splits the pot i ways. w is the event that we win. Lastly E is the expected value of a single play (including splitting the pot and the cost of the ticket).

I don't know the probability that I win along with exactly one, or two, or any specific number of tickets... so to get to the probability I need to get a little creative. I do know the probability that exactly i tickets win given that there is a win.

This can be derived through this thought process. Keep in mind that the probability of i tickets winning givenT purchased tickets and the probability of a single ticket winning P(w) is

Now we want to know the probability that our single ticket wins AND only one ticket wins. First we need to know the probability of a specific number of tickets, i, winning given that there is a win:

In the chart where we saw the probability of splitting a winning ticket i ways, that is the math that was necessary to do it. I just chose an i (starting at 1) and plugged in every value for the expected jackpot that I want to plot. From the analysis I have above (and in the Excel file I link to below) I have a model that predicts the number of tickets purchased for a lottery with a given jackpot size. There are technically an infinite amount of things to plot so you need to just pick a range for i and J. You may need to play with it a bit to find something that suits your liking.

With that portion of the puzzle in place, we can write the following to represent our single ticket winning and there being exactly i winning tickets.

Now we need to convert this into a dollar amount. We just multiply this probability by the jackpot amount spliti ways. To get the total expected value we just need to add all of these up like so:

Translated to the common tongue this basically says that the expected value is the expected value of winning the jackpot and not sharing it plus the expected value of winning the jackpot and sharing it with one person plus the expected value of winning the jackpot and sharing it with two people... all the way to infinity and then all that minus \$2 for the ticket cost.

This simplifies to

Adding an infinite number of things can be difficult or require some more rigorous math. Instead we can approximate E by only adding from 1 to 20.

## Next Steps

From here we could take the data we are now generating in Excel to fit a curve to and come out of this with a mathematical model. Technically I've done this with Excel and arrived at the following:

The problem here is that the equation Excel has come up with doesn't seem to match the data it was fit to. I'm linking to the Excel file I've built in case anyone wants to double check my work for inaccuracies or to see what's up with this issue.

9 responses
I think your analysis is inaccurate. As I understand the structure of the lottery, they return 50% of their revenue as prize money. Since you aren't discounting the jackpot for the time value of money, then you should never have a result of less than \$1 dollar for the expected value of a ticket. For starters you have to take into account the value and the odds of all of the other prizes that can be won. Secondly, you don't seem to have an accurate prize value for the jackpot vs the number of tickets sold. You have a jackpot of \$675 million for 1/9/16 drawing, but the actual value was \$949 million. The conclusion that the expected value of the lottery ticket is still less than the cost is almost certainly correct. The problem is that each ticket sold pushes the expected value back toward 50 cents.
Thanks for the awesome thought-out reply JJ! First of all, to your point on data quality: I purposefully took the first announced jackpot amount rather than what the jackpot ended up being. This was my attempt to remove the influence of the number of tickets bought for a specific drawing influencing my prediction of the number of tickets bought if that makes sense. If you look in the link that is the source of my data you will see the \$675m figure on that date but crossed out. I take that to mean it was updated. You're right that this is all about the expected value of the jackpot only. I may do the rest of the analysis at another point or someone could take the analysis I have from this point and run with it. The way I understand the lottery is that one wins the full amount and can opt for a lump sum with around 50% tax or annuity payments. Annuity payments I think are more like 20-30% but I'm not sure, and you're right that I simplified the analysis by not considering it. I left it out because there are so many options when it comes to tax breaks that I just dropped it entirely. To your point that I should never have an expected value less than a dollar, I'm not sure exactly why you think that. To be sure I'm clear, I am subtracting the cost of playing from the expected value which (if you accept lotteries are not wise investments) must surely be a negative value and less than \$1. Also why does each ticket sold push the value towards \$0.50?
Sorry, with powerball costing \$2 the expected value of a ticket should be \$1, since they are returning 50% in prize money. After many drawings with no winner you would expect the expected value of a ticket to increase, since the jackpot represents a significant amount of dead money at that point. It is true that your expected value goes up as there is more and more dead money, but it doesn't go up as fast as many might expect. The reason for this is that as the jackpot increases, ticket sales increase at an increasing rate (at least for all of the data points that we have). All of the new money coming in reduces proportion of the jackpot that is dead money, thus restraining the growth of expected value. If the prior jackpot were \$10,000,000 and only 1 ticket were sold the expected value of the ticket would be \$10,000,000+ a little bit times the odds of winning the jackpot plus the odds of winning all other possible prizes. But if 10 billion tickets are sold, the jackpot goes up by a lot more, but the dead money is now a smaller proportion of the jackpot which pushes the expected value towards 50% of the price of a ticket. If at some point the ticket sales stop increasing at an increasing rate as the jackpot goes up, that is when the expected value will really start rise.
Your use of expected value is incorrect and shows a disconnect between your understanding of statistics and the layman's perspective. The incomplete equations and graphs are rudimentary and good enough for back-of-the-envelope calculations -- from a "bank" perspective, but you don't include tax burdens or a secondary winnings structure. At no point has playing "lotto" ever approached anything resembling a "real investment". Please. Just. Stop.