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PowerBall EV and Odds Calculator, or, Why PowerBall is a Ripoff

November 24th, 2012 · 2 Comments

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Preamble

As a semi-professional gambler, I feel it’s my duty to explain to everyone why PowerBall jackpot hysteria is largely worth ignoring. A basic understanding of probability and expected value is essential to understand the content of this post, but I’ll try to keep it simple. The calculations in this post are based on the official odds and jackpot values posted on the “how to play” page on the official PowerBall website.

PowerBall in a nutshell

As of 2012, PowerBall costs $2 to play. Players select 5 unique numbers between 1 and 59, and one “PowerBall” number between 1 and 35. The numbers are randomly selected in a drawing twice a week. A player wins when their ticket either matches the PowerBall number, or matches 3, 4, or 5 regular numbers. Matching all 5 numbers, plus the PowerBall number, pays the jackpot. The jackpot starts at $40,000,000. If a drawing does not result in a jackpot winner, the jackpot progressively increases until a jackpot is won. In the event there is more than one jackpot winner, the winners split the jackpot and each receive an equal portion.

There is also a sidebet option called “PowerPlay” that costs $1. The PowerPlay bet increases all the non-jackpot prizes. Players must play PowerBall for $2 to make the PlayerPlay sidebet on that ticket for an additional $1.

PowerBall and other mega lotteries are interesting in the context of gambling psychology, mainly because of the large progressive jackpots that are essentially unparalleled in any other game of chance. Truly, winning a PowerBall jackpot is a life changing event even at the minimum jackpot. In this article, we demonstrate that despite the excitement the game creates, the player has dismal statistical expectations when playing PowerBall.

PowerBall Odds

As noted above, these odds are based on those published on the official PowerBall website. There are nine ways to win:

Match Odds Pays
5 Numbers + PowerBall 1 in 175,223,510.00 Jackpot
5 Numbers 1 in 5,153,632.65 $1,000,000
4 Numbers + PowerBall 1 in 648,975.96 $10,000
4 Numbers 1 in 19,087.53 $100
3 Numbers + PowerBall 1 in 12,244.83 $100
3 Numbers 1 in 360.14 $7
2 Numbers + PowerBall 1 in 706.43 $7
1 Numbers + PowerBall 1 in 110.81 $4
PowerBall and 0 Numbers 1 in 55.41 $4

PowerBall EV (Expected Value)

If you’re here, you probably already know the definition of “expected value” as a term of art in probability analysis, but in case you don’t, check out the Wikipedia article on expected value. Based on the $40,000,000 minimum jackpot, here are the expected values for PowerBall:

Match Odds Pays Weighted Value
5 Numbers + PowerBall 1 in 175,223,510.00 $40,000,000 $.22828
5 Numbers 1 in 5,153,632.65 $1,000,000 $.19404
4 Numbers + PowerBall 1 in 648,975.96 $10,000 $.01541
4 Numbers 1 in 19,087.53 $100 $.00524
3 Numbers + PowerBall 1 in 12,244.83 $100 $.00816
3 Numbers 1 in 360.14 $7 $.01943
2 Numbers + PowerBall 1 in 706.43 $7 $.00991
1 Numbers + PowerBall 1 in 110.81 $4 $.03610
PowerBall and 0 Numbers 1 in 55.41 $4 $.07219
Cumulative Odds 1 in 31.85 Expected Value ($) $.5887
Expected Value (%) 29.43%

You’re reading that right: a whopping 29.43% EV. For the uninitiated, casino regulators in the states of New Jersey, Mississippi, and Nevada set a statutory minimum EV of 83%, 80%, and 75% respectively on slot play. In other words, even playing the worst legal slot machines play at a significantly higher EV than playing PowerBall at the minimum jackpot. For video poker players, instead of slot players, this would be like playing on a machine that normally on high hands, but paid $0 on hands lower than a flush (i.e. ZERO pay on straight, three of a kind, two pair, jacks or better, etc.). Obviously no one in their right mind would play that game!

Of course, the PowerBall jackpot progressively rises if no one wins the jackpot, which brings us to the the meat of this article.

PowerBall EV Calculator

The advertised jackpot for PowerBall as of the date this article was last updated, November 26, 2012, is $425 million, so that is the default value in the calculator. Of course, this is not the true jackpot, instead only a grand total of annuity payments that would be received over a 20-year period. The actual cash value of that jackpot is $278.3 million. Without getting into why annuities are generally terrible investments, the basic idea is that there is a “time value” of money. In this case, in order for the EV of the game to be properly calculated, we assume that the player takes the lump sum. The player can always buy their own annuity, another great reason to always take the lump sum if you win a lottery.

Match Odds Pays Weighted Value
5 Numbers + PowerBall 1 in 175,223,510.00 Enter jackpot cash value in millions of $: $1.5883
5 Numbers 1 in 5,153,632.65 $1,000,000 $.19404
4 Numbers + PowerBall 1 in 648,975.96 $10,000 $.01541
4 Numbers 1 in 19,087.53 $100 $.00524
3 Numbers + PowerBall 1 in 12,244.83 $100 $.00816
3 Numbers 1 in 360.14 $7 $.01943
2 Numbers + PowerBall 1 in 706.43 $7 $.00991
1 Numbers + PowerBall 1 in 110.81 $4 $.03610
PowerBall and 0 Numbers 1 in 55.41 $4 $.07219
  Expected Value ($) $1.949
Expected Value (%) 97.44%

What about the PowerPlay?

The PowerPlay is ripoff just like the game itself. In the context of the otherwise astronomical odds of the PowerBall, the PowerPlay wager serves as a sort of insurance against a jackpot. The PowerPlay bet increases the payout of the more likely winning combinations. This makes weighted value of the lowest prize levels increase relative to the additional $1 wagered, but not enough to justify the additional $1 wager (not even close). Our analysis below shows that the additional $1 wagered has an EV of only 49.95 cents ($.4995, or 49.95% on the additional dollar). Since the PowerPlay doesn’t affect the jackpot, there is no point in providing a calculator for PowerPlay wagers. Just do your self a favor and don’t bother making this bet. The pay table below is for the minimum jackpot of $40 million. Playing the PowerPlay actually reduces the overall expected value of the ticket to a staggeringly low 36.27%! Remember, to play the PowerPlay raises the overall wager to $3, and the EV% we calculate below reflects that.

Match Odds Pays Weighted Value
5 Numbers + PowerBall 1 in 175,223,510.00 $40,000,000 $.22828
5 Numbers 1 in 5,153,632.65 $2,000,000 $.38806
4 Numbers + PowerBall 1 in 648,975.96 $40,000 $.06163
4 Numbers 1 in 19,087.53 $200 $.01048
3 Numbers + PowerBall 1 in 12,244.83 $200 $.01633
3 Numbers 1 in 360.14 $14 $.03887
2 Numbers + PowerBall 1 in 706.43 $14 $.01982
1 Numbers + PowerBall 1 in 110.81 $12 $.10829
PowerBall and 0 Numbers 1 in 55.41 $12 $.21657
  Expected Value ($) $1.08836
  Expected Value (%) 36.28%

This is somewhat counter-intuitive from what I’ve gathered discussing this with people that lack formal education in probability and statistics. Even though the $1 wagered triples what is paid out on the PowerBall alone, and the PowerBall plus 1 number, and even though those two are the most frequent ways to win, the reality is that both events still nearly 40:1 and 100:1 respectively. The reason the PowerPlay wager is, overall, not a good wager, is the PowerPlay is per ticket, and so the player would have been better off simply taking that $1 they’d otherwise wager on each ticket, and simply buy more tickets. Of course as I’ve been describing here, PowerBall is a terrible value in the first place. Players are better off going to the dollar store (but I digress).

What about Taxes?

Death and taxes, the only two certainties in life. Indeed, PowerBall winners are taxed to the hilt. Jackpot winners are immediately welcomed to the highest tax bracket (I hear the IRS even sends you a hand-written thank you when you’re all paid up). Generally winners will face a 25% federal tax, and a 6% to 9% state tax. I live in New York, which prides itself on rampant taxation, so the calculations below use a 34% tax rate. As you can see, this further diminishes the potential return of playing PowerBall. Using the starting jackpot of $40 million (try it yourself), the expected value drops to $.44, or roughly 22% EV! It is difficult to convey in words how abysmal those kinds of returns are for games of chance involving cash wagers.

Enter a tax Rate (use decimal, i.e. 34% = .34)

Match Odds Pays Weighted Value
5 Numbers + PowerBall 1 in 175,223,510.00 Enter jackpot cash value in millions of $: $.1507
5 Numbers 1 in 5,153,632.65 $1,000,000 $.12807
4 Numbers + PowerBall 1 in 648,975.96 $10,000 $.01017
4 Numbers 1 in 19,087.53 $100 $.00524
3 Numbers + PowerBall 1 in 12,244.83 $100 $.00816
3 Numbers 1 in 360.14 $7 $.01943
2 Numbers + PowerBall 1 in 706.43 $7 $.00991
1 Numbers + PowerBall 1 in 110.81 $4 $.03610
PowerBall and 0 Numbers 1 in 55.41 $4 $.07219
  Expected Value ($) $.44
Expected Value (%) 22%

Note that this calculator assumes that any prize under $600 is not taxed. Technically, a winning of any dollar amount, even lowest prize which is just $4, is taxable income. Of course, no paperwork is filed for these types of winnings, as they are redeemed at the retailer, so literally no one reports them or even thinks to around tax time, even though they’re really required to. For the record, taxable winnings can be offset by corresponding taxable losses, so if you are going to make a habit of playing lottery games, make a habit of saving your losing tickets! You can throw them away at the end of the year if you don’t hit a jackpot, but if you happen to, you’re allowed to write-off those losses you accumulated along the way.

The takeaway here is that, at a paltry 22% EV, playing PowerBall when the progressive jackpot has just reset itself is a particularly bad value proposition.

Musing on Number Selection

Because the winning numbers are selected randomly each week (that is, the selection of numbers are independent events) there is no way to “predict” future numbers based on analysis of previous winning numbers. However, human beings are notoriously bad at selecting random numbers. In fact, there is a psychology to lottery number selection.

Since jackpots are split if there is more than one winner, it is important to select numbers that would not likely be selected by others. Frequently, people play dates such as anniversaries or children’s birthdays, etc. This is actually a bad idea, because there are only 365 possible dates in a year! Lots of people will select dates. If you happen to select all dates, and win, there is a small chance that someone else also randomly selected those same dates. Not to say that it wouldn’t be nice to win anyway, but the idea is, since no one can predict the numbers, the best bet is to select numbers that you think no one else will select. This way, on the off chance that you do hit, it’s unlikely that numerous other winners will emerge.

Breakpoint Analysis

Using simple algebra, we calculated that based on the odds and expected values above, PowerBall has a pre-tax EV of $2 (100%) when the cash jackpot is at roughly $287.3 million. Of course the reality is that taxes are a certainty. Factoring in 34% taxes, PowerBall has a post-tax EV of $2 (100%) when the cash jackpot is at roughly $454.2 million. Given that the advertised jackpot is always for the annuity, and based on a rather rough analysis of the relationship between the advertised jackpot and the cash value, that would mean that the advertised PowerBall jackpot would have to be roughly $693.6 million before the true post-tax EV of the ticket was $2 (100%).

To Infinity and Beyond

Even if the expected value is $2, is PowerBall worth playing? Simply, no, it is not. Because the change in the expected value is influenced solely by the jackpot, and since the jackpot is still a 175 million to 1 outcome, it’s extremely unlikely that any player would reasonably be able to ensure they would recoup their wager.

According to the National Weather Service, an individual has a 1 in 10,000 chance of getting struck by lightning in their lifetime. If we assume that lightning strikes are random, that means you are nearly twice as likely to be struck by lightning twice in your lifetime than you are to win PowerBall on any given ticket.

Even more “shocking” is that the National Weather Service estimates that in any given year, a person’s odds of being struck by lightning are about 775,000 to 1. In other words, you’re more likely to be struck by lightning this year than you are to win $10,000 or more at PowerBall.

Before someone tenders a story about someone they know that has been struck by lightning twice, lightning strikes are not in fact randomly distributed. Certain occupations expose workers to higher probabilities of lightning strikes, and geography is also a major factor.

Where do we go from here

I’ve heard people describe playing PowerBall as “paying to dream”. I’m pretty sure you can dream plenty without wagering $2 on a 175 million to 1 chance. You’re far better off taking that $2, or whatever you planned to wager, and buying yourself something that you need or want. If you absolutely must wager that $2, PowerBall is certainly proved here to be a rather foolish way to wager it. Personally I consider slot machines to be a foolish way to wager, but even slot machines pay a far higher return on average than PowerBall.

And finally, what’s with the name? PowerBall sounds like some bullshit made-up sport you’d see in a sci-fi movie. Either that, or some “extreme sports” type of league made up by Vince McMahon. Perhaps the name reminds me of “Pro Thunderball” from Upright Citizens Bridge (below), totally worth watching but having nothing to do with the rest of this article at all:

Tags: My Thoughts · Scary Stuff

2 responses so far ↓

  • 1 Tim Broder // Nov 27, 2012 at 10:39 am

    ahhhh well. I went in on a work pool 2 mins before i saw this

  • 2 Terry // Apr 24, 2013 at 7:50 am

    Great Article. I’m from Australia and Yes we are unfortunate to have Powerball in Australia. They changed the rules of the game in March this year from 5 numbers + 1 powerball to 6 numbers + 1 powerball added another division pool and spun the public it was a good thing as more winners. The odds in winning division 1 has went from 55 Million to 1 to 77 Million to 1. Since March the Jackpot hasn’t went off and is 50 Million this week. The Australian public has been hoodwinked. Your article should be applied to Australia Powerball to assist the public. No tax on winnings. Only lump sums are paid no annuities. I have heard a few Maths intellects in US have cracked some formulas / applied algorithms to predict winning numbers with huge success. Well done on your article.

    [Editor’s reply: You’ve heard wrong. There is no way to predict lottery numbers in these types of drawings. Perhaps you’re thinking of Joan R. Ginther, a statistics professor that figured out an algorithm used in the production of “scratch off” lottery tickets. However, keep in mind, scratch off tickets (and “pull tabs”) are part of a pre-determined wagering pool. Before any player even gets a chance to buy a ticket, the total revenue of the game and the exact dollar amounts of the payoffs are known to the lottery commission. The “randomness” is the distribution of the tickets, not the result.]

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