Wednesday, February 8th, 2012
24

How Much Can You Expect As A Return On That $2 Powerball Ticket?

As has been widely advertised, the jackpot for tonight's Powerball drawing is $250 million. Later today, I'll head out to a store in my Chicago neighborhood to buy a $2 ticket, then spend the rest of the day as I always do before a drawing, daydreaming about what I would do with all that money: A house across the street from Lambeau Field (perhaps attainable without winning the lottery), villas on the beach, bottles of Pappy van Winkle 23-year. The works. Top shelf everything. Living easy.



While I know that my odds of actually winning the jackpot—1 in 175,223,510 to be exact—are essentially zero, I never bothered to calculate what my expected return is. That is, how much money am I expecting to get back after multiplying the odds and the prize money and subtracting out taxes? Spoiler alert: it’s a lot less than $2. In fact, the actual expected return is far worse than I would have guessed.

Because I'm acutely aware that the odds are stacked against me, I’ve made compromises with my lottery purchases. I will only buy tickets when the Mega Millions or Powerball jackpots are over $150 million (anything less would, theoretically, constrain my ability to live the dream); I only buy one ticket at a time; and I only buy tickets with the dimes and nickels that accumulate in between acceptable jackpots. (Needless to say, convenience-store clerks love that last part.) 

In the last year or so, since I've started playing, I've probably bought about 20 tickets—until recently, they were only a dollar.

When I buy a ticket, I choose my own numbers because it makes me feel as though I have some sort of control over the outcome. I roll with the numbers of important Packers; Aaron Rodgers’ 12 or Charles Woodson’s 21 always occupies the moneyball slot. Last week, I picked #4 on my ticket (in the pick-5 category, of course), a sign that I'm ready to forgive Brett Favre for his Vikings transgressions. 



So if the jackpot is $250 million, how much of a return can you expect on a $2.00 ticket? My conclusion was harrowing. After federal and state withholding taxes, my expected return is less than 94 cents. Ouch. Still probably worth my dimes and nickels as it’s not like they have a practical application that extends beyond sitting in a jar on my desk, but it was even less than I thought it’d be. Here's how I figured it out.



METHODOLOGY
The $250 million advertised jackpot is a little misleading. The jackpot is actually an annuity with 30 even payments of “just” $8,333,333.33 over 29 years (the first payment is immediate). The lottery also gives you the option of taking a lump sum payment, which, in this case, would be $156 million. Because only sissies would take the annuity payment, we will use the lump sum as the basis for our calculations.

 The $156 million is immediately subject to a 25% federal withholding tax, which brings us to $117 million. This is now subject to state (and, in New York City and Yonkers, municipal) withholding taxes. To calculate an average, I used the state tax withholding data from USAMega.com.

Because we want to figure out a representative average for the population who might be playing, an arithmetic average of the states’ rates would not fully account for the true proportions that each state contributes. To reconcile this issue, I used the electoral college as a guideline for proportions.

Using 270 To Win's electoral map, I subtracted out two electoral votes from each lottery-eligible state (42 of the 50 states + Washington D.C. have Powerball) to eliminate Senate equality, and calculated a weighted average (see Appendix 1 for Excel formulas) state withholding tax of about 4.6%. 

This now brings the lump sum payout to $111,611,351.58. Via Powerball's website, the odds of winning the jackpot are 1 in 175,223,510. Multiplying the payout and the odds together, we get an expected return of about 64 cents.



However, the jackpot is not the only way to win a prize on a Powerball ticket. This graph from the website shows other prizes that you can win and the odds of winning them:

Using the same methodology from above, the expected returns of the other prizes are, respectively, 14 cents, 1 cent, 1 cent, 1 cent, 2 cents, 1 cent, 4 cents, and 7 cents. Please see below for the Excel macro that details the calculations and places them in a table.

CONCLUSION
Add it all together and your average expected after-tax return on a $2 ticket (bought with after-tax income) is about 94 cents. It’s actually even a little bit lower because if multiple people win the jackpot, the prize is split evenly among them.

Simply put, a Powerball ticket is an absolutely terrible investment, far worse than any casino game. Via Insider LV, the house has a .6% advantage in a six-deck game of blackjack, 1.41% on Pass/Come Craps, and 8.1% on dollar slots.

In Powerball—when the jackpot is much higher than it is normally—you are relinquishing more than 53 cents per dollar in expected return after taxes.

I still don’t know that I have a particularly better use for dimes and nickels, but after this analysis, I’m likely almost completely done spending dollar bills or valuable laundry-eligible quarters in return for little more than the ability to daydream about what I’d do with the winnings.

(Download the Excel spreadsheet and appendices here.)



Ryan Glasspiegel is a freelance writer based in Chicago. He writes Sports Rapport. Follow him on Twitter @RGSpiegel .

24 Comments / Post A Comment

barnhouse (#1,326)

Awesome. What I'd really like to know is this: how many jackpots go unclaimed–and what is the largest one to go unclaimed? Only imagine buying a winning ticket and then forgetting to check the number. A sad old expired bit of paper in a forgotten coat pocket.

stuffisthings (#1,352)

Three posts today, and already two have a chart and/or spreadhseet. Is it my birthday or something?

EDIT to say: Interestingly, the net present value of the annuity at an 8% discount rate is "only" $93 million before taxes, so the lump sum IS a better deal!

My lottery rule is only to play on roadtrips. Jackpots always seem to be especially high around the holidays. (Conspiracy?) Anyhow, I have a mental discipline for not playing more, which is that I force myself to play the combination of 1-2-3-4-5-6. This combination is no less likely than any other to be a winner and yet, who in their right mind would play this combination? The mind is forced to face the conclusion is that no one in their right mind should play any combination. Each and every one choice as foolish as 1-2-3-4-5-6.

djfreshie (#875)

@Charismatic Megafauna
That is the most beautiful bit of reasoning I've ever read.

BardCollege (#2,307)

@Charismatic Megafauna Since you can be certain other people are playing these numbers as well; if you ever do win you will have to share the jackpot with A LOT of people. If you are going to be stupid enough to buy a lotto ticket at least be smart enough to use a random draw with lower odds of sharing the prize.

Louis Fyne (#2,066)

@Charismatic Megafauna That's the combination to my luggage!

stuffisthings (#1,352)

@Charismatic Megafauna I set my password to ********

deepomega (#1,720)

Marvelous. Do getting struck by lightning next!

barnhouse (#1,326)

YES. I want to read that.

@stuffisthings – I calculated the implied discount rate of the annuity to be 3.65%

Because the first payment, $8,333,333.33, is immediate, you subtract it from the $156,000,000 PV. Therefore:
PV = 147,666,666.67
FV = 0
N = 29
PMT = 8,333,333.33

BirdNerd (#4,196)

Love this analysis. A dude at work collects for tickets when the jackpot get high enough. On average there are 35-40 of us and at $2 per person I wonder what our expected return is?
I never play alone, but somehow I feel like sharing all these numbers with these random folks I don't really care for gives me better odds of winning at least something one day.

stuffisthings (#1,352)

@BirdNerd The expected return would be exactly proportional, since your higher chances of a win are offset by having to split the jackpot. The best way to see this: if you bought 175 million tickets (for $175m) you'd be, theoretically, guaranteed to win the $111 million. $111m/$175m = ~0.64. Still a bad deal!

@BirdNerd : More importantly, you don't want to be the one guy in town who didn't buy a ticket.

BirdNerd (#4,196)

@stuffisthings THANKS! I understand now, just needed some hand-holding.

And for the record, I think the lump sum is a better deal than the annuity. The mitigating factor is that very few people are risk seeking enough to play the lottery AND invest the lump sum soundly when they win.

roboloki (#1,724)

that's $6.58 in dog money.

Also, the Don't Pass line offers even better odds, at the price of looking like a contrarian asshole.

There's a moral there somewhere.

BardCollege (#2,307)

The expected value is even worse because you have to also account for the odds that someone else (or god forbid two people!)will win the prize forcing you to share the jackpot with them. Even worser, as the value of the jackpot increases the odds of a shared prize also increase, because more people buy tickets. This makes the math a little harder, but I can guarantee you that it makes the EV even smaller.

whizz_dumb (#10,650)

This is interesting. Also interesting, some of my best friends are Packers fans based in Bear Country.

@BardCollege I noted that in the conclusion/had no idea how to figure out the math. And if I had figured it out, I wouldn't have been able to explain it in words.

iracane (#4,287)

Delaware does indeed have a state income tax. The highest bracket is 5.95%. I am certain this will not affect your final figure.

lawyergay (#220)

The odds of winning are 50-50: You either win or you don't!

joe08smith (#214,879)

cool

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