# Powerball

on January 9, 2016 at 4:00 am

Today’s Powerball lottery offers a prize of \$800 million. Is the prize high enough to make it worth playing for an economist? In other words, is the prize high enough to be a net gain in expected value terms? Almost!

The odds of winning are 1 in 292.2 million. So the expected value of a ticket is \$800*1/292.2=\$2.73. A ticket only costs \$2 so that’s a positive expected value purchase! We do have to make a few adjustments, however. The \$800 million is paid out over 30 years while the \$2 is paid out today. The instant payout is about \$496 million so that makes the expected value 496*1/292.2=\$1.70. We also have to adjust for the possibility that more than one person wins the prize. If you play variants of your birthday or “lucky” numbers that’s a strong possibility. If you let the computer choose your chances are better but with so many people playing it wouldn’t be surprising if two people had the same number–I give it at least 25%. So that knocks your winnings down to \$372 million in expectation.

Finally the government will take at least 40% of your winnings, leaving you with \$223 million in expectation. At a net \$223 million the expected value of a \$2 ticket is about 75 cents. Thus, a Powerball ticket doesn’t have positive net expected value but the net price of \$1.25 is significantly less than the sticker price of \$2. \$1.25 is not much but to get your money’s worth buy early to extend the pleasure of anticipation.

1 Steko January 9, 2016 at 4:58 am

There are some lower payout wins that mitigate the losses a bit as well.

2 gregor January 9, 2016 at 8:34 am

I believe the expected payout of the sub-jackpot winnings is about 30 cents per-tax.

3 Don Meaker January 10, 2016 at 1:55 pm

Of course the 5 minutes of fantasy can be obtained for free, as one decides whether or not to buy a ticket.

The lottery was a better deal when the Mafia ran “The Numbers”. The tickets only cost 50 cents, they put half into the prize fund, and the winners didn’t pay tax on prizes.

By comparison to the CA lottery, only 1/3 goes into the prize fund, and the big winner is government, who gets 1/3rd up front, 1/2 in federal tax and 1/10th in state tax. The rest goes to the crony capitalists, who in exchange for small campaign contributions, get 1/3rd of the winnings for the priviledge of making and selling tickets.

4 Paul January 9, 2016 at 5:29 am

I think expected value does not capture the full utitlity of playing the lottery. I consider myself an economically literate person, I save for retirement, I understand opportunity costs and so on. Still I play the lottery for a small amount every month (less than a percent of my monthly net income). I thought about it this way:

If I save say 5 \$ (well its € here) per month for 30 years with 5% interest and lets assume ticket prices rise 2,5% per year I make about 5-6k € after 30 years. I mean it is a nice sum but in my overall lifetime income stream its very insignificant. I will have saved (or been forced to save via social security) much more money. I will not really miss 5k, this amount of money won’t change my retirement lifestyle much. But winning a a larger amount even if it is only 100k instead of millions would make a noticable difference.

One might see it like this: 5k after 30 years is not a discrete margin. The expected value theory holds up well for bets or casino games but winning the jackpot is more like black swan event. You might want to insure against not being able to capture that event.

5 MG January 9, 2016 at 6:49 am

Adjusting for utility of the reward makes sense, and explains some of the “overvaluation” relative to risk less investments. It would be interesting to value the “weekly for life lottery strategy” against the cost of investing small amounts in riskier payouts with a comparable probability (of course it would be low) of accumulating as much as the 100K (or bigger hurdle, if you wanted). For example, what would it cost to buy short-dated, out of the money puts that occasionally produce large payouts. Of course, here you appreciate another advantage of lotteries — small units of investments, which absent pooling are not achievable in financial markets.

6 Rusty Synapses January 9, 2016 at 7:30 am

I agree with all of this, plus “the pleasure of anticipation” as Alex says is real entertainment value for me (I assume economists don’t think entertainment spending is irrational). I only play when there are these really large jackpots, and it’s fun to daydream about what I’d do if I won (and the money I spend is completely insignificant).

7 Slocum January 9, 2016 at 8:36 am

I do understand the small pleasure of buying a ‘dream ticket’, but I’m disgusted by the way that lotteries exploit people. State lotteries return only 2/3 of the monies collected as prizes — a rate that far worse than at casinos. And then the big prizes are taxed at a high rate (while very few losing tickets are ever taken as tax deductions), so knock down the prize money by another 40%, and you end up with the government taking a 60% share and the winner getting just 40. Add in the forcible suppression of competitors who offer a better deal (via SWAT raids when ‘necessary’) and the ubiquitous slick advertising campaigns to induce vulnerable people to buy tickets, and it’s all really pretty sickening.

8 Mike January 9, 2016 at 10:10 am

Nice post, agreed. Also, my impression is that lotteries are much more popular among lower-income folk.

9 BC January 9, 2016 at 2:15 pm

Government monopoly on gambling has a disparate impact on the poor.

10 Alan Gunn January 9, 2016 at 11:05 am

Yes, and for some people it’s even better than that. There was a study some years back (sorry, don’t have a citation) showing that a lot of lottery players are people in early middle age who hate their jobs but have no way to retire early. The extra beer or two or movie each week that they’d buy if they didn’t play the lottery would mean nothing much. To be sure, there are also studies showing that people who win big bucks in the lottery are no happier than anyone else, and no less likely to go bankrupt. And then there are studies concluding that the results of most studies are wrong.

11 T. Shaw January 9, 2016 at 12:00 pm

Good points. My nonfinancial brother calls large lottery prizes “investment grade” when the nominal prize is above the odds for the jackpot.

I ensure that my charitable contributions far exceed lottery outlays.

My case, I don’t spend a lot of money on beer (one jigger Dewars and two cheap cans of beer a night, whether I need it or not), movies (it’ll be on TV), plays (fall asleep), eating out (only when I have to). I have comparatively high disposable income: substantial annuity and investment income, small, low-rate mortgage (have cash to repay), the rental property carries itself, the kids’ college costs are paid, etc.

And, I have plans for the jackpot money when I win. A former colleague was derisive of my playing, but admitted he is interested in how a financial pro would handle such a windfall.

12 T. Shaw January 9, 2016 at 11:45 am

I have \$10 in tonight’s (now) forecast \$900 million jackpot. I had about that much in last night’s \$165 million Mega Millions jackpot. More interesting, the Warden, who never plays, has \$8 in it. The other day, I saw a man buy \$100, or 50 tries. I imagine it’s a pool or, maybe, it’s his \$100.

I play various, big-prize games. I understand that winning a big jackpot would be a major, massive miracle. However, there is some entertainment value in that, for a few hours, I can imagine, dream if you will, winning and all that I would do with the money. That’s more like behavioral economics.

Last night a player in Staten Island (Richmond County), NY won the Mega Millions nominal \$165 million. Someone wins. It might as well be me.

Of course, if you’re throwing away the rent money or Little Johnny won’t have a new pair of shoes, it’s a no-go.

I’m an optimist. I save the losing tickets so the losses can be offset against gambling winnings for income taxes . . . when I win the big one. That’s the optimistic accountant in me.

13 wiki January 9, 2016 at 6:03 pm

Reuven Brenner wrote several books three decades ago in which he modified risk analysis to take into account exactly these phenomena (and he specifically mentions lotteries). Cf. Betting on Ideas, 1985 as well as subsequent books.

14 Jan January 9, 2016 at 5:33 am

Thanks for the running the numbers. I don’t play the lottery, but it’s kind of crazy that the expected value of a ticket in this case is anywhere close to what it actually costs. Incidentally, my 92 year old neighbor, first person to move into my neighborhood in the 50s, stopped by to drop off a ticket for my family yesterday.

15 Cliff January 9, 2016 at 10:54 am

The lottery normally returns around 50% so it’s not crazy at all.

16 TMC January 9, 2016 at 11:47 am

“stopped by to drop off a ticket for my family yesterday” Hard to find neighbors like that anymore. Maybe ever.

17 BC January 9, 2016 at 2:26 pm

I think casinos typically payout over 90 cents per dollar. That’s the difference between a government monopoly and private competition.

18 Shamus January 10, 2016 at 7:02 pm

House odds vary by game. The worst are on the so-called fortune wheels and the best are on Blackjack. I’ve seen house odds on Blackjack quoted as low as 2%.

19 Sbard January 11, 2016 at 5:53 pm

Craps is pretty close (<2%) and doesn't require any skill. Just bet the pass or don't pass line (don't pass is a very slightly better bet) with full odds. No decisions to make.

20 WC Varones January 9, 2016 at 7:27 am

You can estimate the number of tickets sold for this draw by the change in the jackpot from last draw to this draw.

If half of every \$2 ticket goes to the jackpot (I think it’s actually somewhat less than half), that would mean the number of tickets sold equals the increase in the jackpot.

I believe the odds you’d split the jackpot are WAY bigger than 25%.

Remember, you’re the marginal hypothetical player here, so if 1 person actually wins, that means you would have split it with him; if 2 win, you would have split 3 ways, etc.

21 Rusty Synapses January 9, 2016 at 7:34 am

Yes, splitting odds are way bigger. Remember in your basic probability class when you figured out how many people you needed in a group before the odds of sharing a birthday were over 50%? If I remember, it’s something around 20, not 365/2, Although as noted, it really matters whether you pick numbers people are likely to pick (days are 31 or less, months are 12 or less, etc.) – that’s why I do quick picks.

22 Anon January 9, 2016 at 9:01 am

24.

23 too hot for MR January 9, 2016 at 9:42 am

It seems to me that if you know there are common picks, you’d outperform quick pick EV by avoiding those numbers altogether and picking quasi-randomly from the rest.

(In addition to sub-32, you’d escape a lot of splits with numerology types by avoiding the digits 7 & 8.)

24 Excursive January 9, 2016 at 11:37 am

Buy 1,2,3,4,5,6. It’s as likely as anything else, and no one else will.

25 Mark Thorson January 9, 2016 at 12:01 pm

No, buy the Fibonacci, Lucas, and triangle numbers. When you win, go on TV and say “The lottery is fixed! Here is the formula I used to calculate the winning numbers!”.

26 Das January 9, 2016 at 12:04 pm

A nice thought that you share with about 10k others…

27 RustySynapses January 9, 2016 at 3:05 pm

Agree with comments on 1,2,3,4,5,6 and avoiding sub-32 altogether. I know @MarkThorson is joking, but if you win the lottery, going on TV and saying it’s fixed, while fun, may not be smartest next step (EV = large negative (goodbye winnings, hello prison)).

28 TMC January 9, 2016 at 11:49 am

This is how I pick horses at night at the races. Winning number is random, so I try to contol the payout by picking a horse with a stupid name, or just follow the shorter line.

29 gregor January 9, 2016 at 8:45 am

The odds of X number of winners is binomial with the trials being the number of tickets and the probability being the 1 in 292 M. This is essentially Poisson with mean equal to tickets/292M. Alex’s discount factor isn’t unreasonable I don’t think.

30 WC Varones January 9, 2016 at 9:12 am

Gregor,

No. There’s a flaw in your logic.

As of now, the jackpot is up \$300 million since the last draw. If roughly half of ticket sales go to the jackpot (a generous stipulation), that means there is likely already one or more winners (mean slightly higher than one).

If you now assume that you also have a winning ticket, you have to split the jackpot with the (mean > 1) other winners.

31 WC Varones January 9, 2016 at 9:15 am

In other words, the probability of one of the 300 million+ tickets already sold winning is independent of your assumption that you also buy a winning ticket.

32 gregor January 9, 2016 at 11:48 am

I said X winners, including zero in the domain of X. That’s the starting point for the problem. Suppose they sell 400M tickets. Then they draw the winning combination from among the possible 292M combinations. What are the odds of no winner by your calculations? How is that not a binomial distribution?

At 400M I get (approximately)
P(0)=25.4%
P(1)=34.8%
P(2)=23.8%
P(3)=10.9%
etc.

To condition on having a winner, divide through by 1-P(0)

1 way split, 47%
2 way split, 32%
3 way split, 15%
etc.

So 47%*1+32%*(1/2)+…

I get an expectation of about 69%. Alex’s 75% discount is not outrageous and seems consistent with about 320M outstanding tickets.

Now I see your point that given that they’ve sold say 400M tickets already there’s about a 75% chance that the winning ticket is already out there, so you’re saying oh there’s a 75% chance I’d split my ticket if I won. But actually not because when I jump in making it 400M+1 there’s STILL a 75% (plus epsilon) chance the winning ticket is out there, and my ticket is just as good as any other. I think the paradox is that the probability of a win is so remote and the effect of one more ticket is so small.

33 WC Varones January 9, 2016 at 12:07 pm

No, you’re still missing it.

We’re only talking about the case where it’s a given that you have won. And in that case you do not reduce the likely number of other winning tickets by one; they are independent of each other.

34 WC Varones January 9, 2016 at 12:13 pm

Think of it this way:

Assume tonight you do not play and there is one winning ticket (the most likely scenario if 300-400 million tickets are sold). Now you get a time machine and get to go back and buy the winning numbers. Your having purchased the winning numbers doesn’t mean the other guy doesn’t still have the winning numbers.

35 gregor January 9, 2016 at 3:29 pm

Given that someone hits the jackpot, what is that person’s expected share of that jackpot?

36 WC Varones January 9, 2016 at 3:40 pm

1/(X+1).

You’re increasing the expected number of winners by one when you assume one specific winner.

37 gregor January 9, 2016 at 5:04 pm

Okay, you are saying add 1 to all my X’s above so that my P(0) becomes P(1) and so on. In that case I get a 54.5% expected share with 400M tickets and 62.5% with 300M. Alex’s 75% would correspond to about 175M tickets.

38 WC Varones January 9, 2016 at 6:04 pm

I think you want to do

100% * P(0) + 50% * P(1) + 33.3% * P(2) + …

39 dan January 9, 2016 at 7:37 am

see oster (2004) on regressivity of powerball in CT (http://www.ntanet.org/NTJ/57/2/ntj-v57n02p179-87-are-all-lotteries-regressive.pdf), which specifically states “the tax is progressive when the income
elasticity is equal to one, in this case at a jackpot of \$806 million”

of course that was in 2004 — but maybe it’s time to update and see how the prediction holds up now that we have a jackpot of that magnitude.

40 Rich berger January 9, 2016 at 7:39 am

I am surprised that an economist didn’t take into account the opportunity cost of waiting in line to buy a ticket.

41 Jeff R. January 9, 2016 at 9:14 am

+1

42 AnthonyGonzalez27780 January 9, 2016 at 9:55 am

At my supermarket, they place the lottery ticket counter right behind the checkout lanes so you pass it on the way out.

43 Sbard January 11, 2016 at 5:55 pm

At my supermarket, you can just buy your ticket from a vending machine, no line.

44 Mark Thorson January 9, 2016 at 12:03 pm

An economist weighs the missed opportunity cost of the labor and capital which could have been used to buy a bag of potato chips. And those were good potato chips indeed! I’m savoring every one!

45 Steve January 9, 2016 at 7:42 am

I find waiting in line for a \$2 ticket watching those in rags in front of me spending \$20-40 on tickets depressing. Isn’t that a psychic cost, or something?

46 Jonathan January 9, 2016 at 8:38 am

Gambling losses are tax deductible to the extent of gambling winnings, so if you get a thrill from gambling you can mitigate some of the tax bite by spending the rest of year by losing at blackjack or poker.

47 Art Deco January 9, 2016 at 11:27 am

Poker is a social activity. Buying lottery tickets is not, unless you gab with other ticket-buyers about their bogus systems.

48 T. Shaw January 9, 2016 at 12:56 pm

I see the same old geezers every time I buy tickets. If one isn’t there I worry about him/her.

49 Art Deco January 9, 2016 at 2:57 pm

The geezers shouldn’t be spending their Social Security on that. They should be playing dominoes, bridge, poker, or mah johngg.

50 Hypo January 9, 2016 at 8:44 am

Decreasing marginal utility of wealth is one of the justifications for high and progressive taxation by the government.

Yet the government runs lotteries that take a little from a lot of poor and middle class people to make one rich person.

And now does it to make one super duper rich person.

And they say this stuff is humorless.

51 anon January 9, 2016 at 9:23 am

+1

52 TMC January 9, 2016 at 11:54 am

It also makes out in that now that one person is pay half of that in taxes, where as the inputs were after taxes.

An increase in collection for no real economic activity at all.

53 Helmhotlz January 10, 2016 at 12:29 pm

But a completely voluntary one entered into by both consenting parties with rules laid out beforehand etc.

54 Curt F. January 9, 2016 at 8:46 am

You also have to take into account that the people running the lottery may simply be trying to scam you. https://www.washingtonpost.com/news/post-nation/wp/2016/01/09/the-company-that-runs-powerball-had-a-16-5-million-jackpot-rigged-by-a-former-employee/ Although because bigger jackpots attract more attention and thus “sunligh”, I suppose the chance of a lottery employee swindling you go *down* as the jackpot goes *up*.

55 Gochujang January 9, 2016 at 8:54 am

I think it is fine to buy a ticket or two, especially if the opportunity benefit is skipping a soda or a cigarette. That said, my sense of loss is greater than my sense of anticipation, so I don’t.

56 RPLong January 9, 2016 at 9:11 am

Should any my fellow commentators win, I would like you to remember as you plan out what to do with your winnings that I believed in you even when Alex Tabarrok didn’t.

57 edmeasure January 9, 2016 at 9:16 am

Expected profit is a very naive way of calculating utility. What you really want to calculate is the expectation of increasing future copies of your genes, which almost certainly depends in a non-linear way on your income.

58 too hot for MR January 9, 2016 at 10:19 am

Fertility and income are negatively correlated, and everybody here knows that.

I can’t discern a coherent point from your comment, unless it’s a joke that nobody wants to screw the sort of loser who plays lotto.

59 Alain January 9, 2016 at 11:36 am

Is that true thoughout the income/wealth curve? It sure doesn’t seem like it from anecdotal data. It is true that as the population moves from poverty to middle class fertility goes down, but a random sampling of billionaires suggests that they have many children.

60 too hot for MR January 9, 2016 at 12:11 pm

Perhaps it’s a u-shaped curve, with hood rats and billionaires doing overtime breeding. Of course one end of that curve is more heavily populated than the other.

61 chuck martel January 9, 2016 at 12:24 pm

“fertility goes down”

That’s a common misuse of the term “fertility”. Instead the statement should be that “reproduction goes down”. The two things aren’t the same. Fertile organisms may or may not reproduce.

62 oh geez January 9, 2016 at 12:44 pm

Chuck, the entirety of social sciences have agreed on the use of “fertility rates,” whatever your hangups as an armchair biologist may be.

63 chuck martel January 9, 2016 at 2:20 pm

The entirety of social sciences have agreed on all sorts of fallacies. This just one more.

64 edmeasure January 9, 2016 at 2:01 pm

Let me try an analogy. Our brains were wired for an earlier world.

Something like 8% of the population of a big chunk of Asia is descended from Ghengiz Khan. Historically, being rich gave one’s children a much better chance of reaching adulthood. If, as I hypothesize, the reward is a non-linear function of wealth, then Alex’s
analysis breaks down.

65 Dr. Diggler January 9, 2016 at 3:00 pm

“The only thing standing between me and indiscriminate f—ing is a \$900m lotto payout.” Seems legit.

66 almond January 11, 2016 at 9:47 pm

This new religion is even stupider than the last one. God isn’t going to make you happy and neither are billions of hypothetical decedents.

67 Thin-Skinned Masta-Beta January 9, 2016 at 9:23 am

Lotteries:

A tax on stupidity and greed when I’m not playing.

A highly leveraged high risk investment when I do play.

As pleasurable as the dreaming and anticipation is, I fear it’s narcotic-like a distraction that prevents energies being mobilized for more productive endeavors.

68 Thin-Skinned Masta-Beta January 9, 2016 at 9:26 am

Hey can anybody do the math on how “irrational” it is to buy the Power Play® upgrade for \$1 in addition to the \$2 base ticket?

Power Play® is a special add-on feature that allows a winner to increase the original prize amounts (except the jackpopt). The Power Play option costs \$1 and must be selected when you buy your Powerball ticket. Powerball players with the Power Play option winning any prize other than the jackpot will win a larger cash prize. The Match 5+0 prize with Power Play option grows to \$2,000,000 paid in a cash lump sum (no annuity option is offered). All other lower prizes (below Match 5) will be multiplied by a number that we draw (2X, 3X, 4X, or 5X). In October 2015, a 10X multiplier was added to the mix with a chance to be drawn for all jackopts at \$150 million or less. As with any lottery prize, if an extremely popular number is drawn and there is not enough money in the prize pools or reserves to cover it, we may need to reduce the prize.

69 Yancey Ward January 9, 2016 at 9:43 am

So, Alex, how many tickets did you actually buy?

70 Art Deco January 9, 2016 at 11:25 am

No market for Lottery tickets in Fairfax City.

71 Bill January 9, 2016 at 9:45 am

I read all of the posts above, and noticed that no one mentioned

the utility

of IMAGINING you have won when you purchase the ticket and what you will do with your winnings.

Something goes off in your brain that is an immediate pleasure award just from purchasing and IMAGINING,

Just like when you purchased that the Playboy magazine before they changed their photo editorial policy.

72 Thomas January 10, 2016 at 1:04 am

You can imagine that you’ve won without purchasing a ticket. In fact, my wife and I already have. Great stories. She wants kids. If we win 800m on our ticket we haven’t purchased, we’ll have 10 children (possibly with surrogates) and invest 10m + 50k for each. 50k + growth to be received at 18 for school, and 10m + growth to be received at age 30 in monthly payments. We’ll pay off our parents homes, buy nice (500 – 1000k) homes in the city of each parents’ for visiting, and buy a \$20m estate, and maybe another \$10m vacation home. We’ll guarantee her admission to Yale Law with a sizeable donation, and regardless, live off of a 3% return of the remaining value, 70% in index funds, 30% subject to directed investments (as our new joint primary profession).

Wow, and I didn’t even have to spend \$2!

73 WC Varones January 9, 2016 at 9:51 am

Now if you could put together a syndicate that was somehow operationally able to buy all 292 million combinations, it comes close to being a winning bet, because you can offset your taxes with deductions from all the losing tickets. However, with the very high likelihood of at least one other winner to split the prize with, it’s still probably a negative EV unless the CASH payout goes over \$600 million.

Related ruminations on Mega Millions here: http://www.wcvarones.com/2012/03/dont-buy-mega-millions-tickets.html

74 chuck martel January 9, 2016 at 10:11 am

Playing a big Pick 6 carry-over at the track makes a lot more sense than the mindless purchase of a lottery ticket. The horse player also doesn’t slow up the purchase of gasoline, cigarettes and soda for the non-idiots waiting in line.

75 WC Varones January 9, 2016 at 10:36 am

“the purchase of gasoline, cigarettes and soda for the non-idiots”

Mmmkay.

76 Glen January 9, 2016 at 10:42 am

Alex says the “instant payout” is only \$496 million, so he works with that amount. But what about the other \$304 million? If it’s paid out in smaller amounts over many years, shouldn’t its discounted present value be added to the \$496 million?

77 Cliff January 9, 2016 at 10:58 am

The other \$304M doesn’t get paid out at all if you choose the instant payout.

78 Ryan Cousineau January 9, 2016 at 10:50 am

The cheaper alternative, once suggested by Tim Harford, is to short the lottery: pick six numbers, never buy a ticket. All the excitement of playing the lottery, but with a +EV!

79 Gochujang January 9, 2016 at 11:00 am

Really? You face the small but very small chance of those numbers coming up, and the pain.

Better, try it yourself.

Flip a coin a million times. All heads? You win the lottery!

80 Cliff January 10, 2016 at 12:39 pm

Your odds of winning the lottery are way, way better than flipping a million heads

81 T. Shaw January 10, 2016 at 2:26 pm

Consider that when one buys a ticket, one is not really paying for a chance at winning — which is astronomically minuscule — but for a few minutes of fantasy about winning. So, at 5 minutes of fantasy for \$2.00, that’s 40 cents per minute. Contrast that with phone-sex rates and it’s a pretty good deal. . . .

82 Bill January 9, 2016 at 11:54 am

Yes!

Another way you play this game is to write down the numbers you were going to bet, put the money in an envelope, give the envelope to your spouse, telling her to give you the money from the envelope if YOU LOSE or to set it aside for a big item at the end of the year. You will be rewarded.

83 Thomas January 10, 2016 at 1:07 am

I like this!

84 Skip Garibaldi January 9, 2016 at 11:06 am

Even if it were true that the expected value of a Powerball ticket was greater than the cost (after counting taxes and the possibility of splitting the jackpot), it still would not be a good investment to buy a ticket, at least if you believe in mean-variance portfolio analysis. See http://www.maa.org/programs/maa-awards/writing-awards/finding-good-bets-in-the-lottery-and-why-you-shouldnt-take-them

By the way, it has never happened for Powerball or Mega Millions that the expected value was greater than the face value. But sometimes it has happened for particular drawings for state-specific lotteries, like Lotto Texas.

That only addresses whether or not you will earn any money. I personally buy lottery tickets (a few dollars a year), although with an expectation of losing all of that money. I also buy candy bars, knowing I will never get that money back either.

85 Excursive January 9, 2016 at 11:45 am

If you’re willing to buy 1,2,3,4,5,6 then go to it. That single example illustrates how bad we are at understanding what 292 million possible outcomes means. If you’re not willing to buy 1,2,3,4,5,6 then don’t play.

86 Excursive January 9, 2016 at 11:49 am

Or if you think someone else might choose the first six sequential integers because of birthdays, choose the last 6 instead.

87 Chris Renner January 9, 2016 at 11:53 am

I’m surprised no one’s questioned the premise that an instantly acquired fortune is a “win”, given the actual history of lottery winners (and of poorly governed countries with suddenly discovered mineral wealth).

88 Thomas January 10, 2016 at 1:09 am

People here (allow me some latitude to speak for others) believe that the history of lottery winners does not accurately predict the result of a median American winning the lottery, given the selection effects.

89 T. Shaw January 9, 2016 at 12:04 pm

The consistent winner in these huge lottery prizes is the government, which will collect hundreds of millions of dollars in personal income tax revenue.

90 Art Deco January 9, 2016 at 2:54 pm

It’s perfectly voluntary, so it’s not a tax. It’s also a minor rivulent in the whole revenue stream. The problem is not that it sells a ticket to some people who amuse themselves that way. The problem is the hard-sell promotions (which you saw little or nothing of in New York 45 years ago) and the people who will go and buy 16 tickets carefully choosing their numbers because they have a ‘system’. There’s a reason AME and NBC Baptist pastors are commonly disgusted with lotteries.

91 T. Shaw January 9, 2016 at 4:22 pm

Truth. Gambling is a crime except when the government runs it. Funny how that works.

Government simply is the criminal activity we consent to do together.

92 Art Deco January 9, 2016 at 5:40 pm

What are you talking about? There are race tracks all over the state, Indian casinos, church bingo, and whatever you can bet on over the internet.

93 Thomas January 10, 2016 at 1:31 am

I know you aren’t that dull. Try not to be misleading or outright lie. You know that gambling is heavily regulated to the point that the only legal examples are government run or single digit per state operations that are government approved.

94 Art Deco January 10, 2016 at 12:16 pm

or single digit per state operations

We get it. You’re innumerate.

95 Ironman January 9, 2016 at 12:11 pm

For anyone interested, we have a tool for calculating both The Odds of Winning and the Magic Jackpot for any Powerball-type lottery, where we can factor in the top federal, state and local income tax rate that might apply for where you live (lottery winnings are treated like ordinary income for federal taxes, but they’re not subject to Medicare or Social Security taxes, or the Obamacare income taxes).

By our math, the minimum jackpot needed to make playing the Powerball worthwhile ranges from a maximum of more than \$1.225 Billion for residents of New York, NY (39.6% federal income tax rate + 8.82% state income tax rate + 3.88% local income tax) to a minimum of a bit over \$965.5 Million in the 9 states without any state income tax or also those states that don’t tax lottery winnings (such as California, in parts of the state without local income taxes).

96 WC Varones January 9, 2016 at 12:34 pm

Ironman,

Good work but I think you are missing one critical input and one less important input.

The critical one is how much of the jackpot is “dead money” rolled over from prior weeks. If most of the money is dead money, you have a much lower chance of having to split the jackpot. If you had \$900 million of live tickets in play, your EV would be very negative because you would expect to share with 3 or so other winners.

In this case if we hit your \$965.5 million I suspect it’s still negative EV because there’s only \$500m dead money so you’d very likely have to split your jackpot with at least one if not two or more people.

The lesser input is the minor prizes, but these vary a lot and aren’t big enough to make a huge difference.

97 WC Varones January 9, 2016 at 12:41 pm

In fact, the EV declines now as the pot gets bigger because there is more live money fighting over a fixed amount of dead money!

98 Enrique January 9, 2016 at 12:58 pm

Go Alex … Lotteries are scams, and if they weren’t state-run monopolies, they would have been shut down long ago under consumer protection laws: http://priorprobability.com/2016/01/09/powerball-is-a-scam/

99 dbp January 9, 2016 at 1:03 pm

One thing not taken into account is that the \$2 price for the ticket does not need to be your net cost. You can write-off gambling losses, so you can discount this just like you would your mortgage interest–if you itemize.

100 Jonathan January 9, 2016 at 5:46 pm

As I said above in regards to this point, losses can only be used to offset gambling wins. So a win reduces the tax cost of future gambling. This increases the net utility from future gambling, but only for the rest of the year unless you choose the 20 year option.

101 Peter Bradshaw January 9, 2016 at 1:04 pm

This [1] old BBC podcast had a fun segment on the relative likelihood of dying between buying a ticket and the draw and the likelihood of winning the jackpot.

Given that they stop purchases a bit before the draw apparently British men over 18 and women over 30 were more likely to die before the draw than win, even if they bought the ticket at the last moment.

[1]http://www.bbc.co.uk/programmes/b0132pk7

102 Thomas January 10, 2016 at 1:33 am

Someone should run ads about this in the states. What are the chances that state or federal authorities would pursue legal or criminal charges against anyone running such ads?

103 Jared January 9, 2016 at 3:05 pm

I’m not sure I understand the multiple winner discount.

If everyone picked their number with a computer, then the expected return for everyone must be the same (496 * 0.6) / 292.2 = \$1.02, because there are no differentiating factors between each person. If other people are picking favorite numbers and you do not, then their odds of sharing the payout go up, and yours go down. Shouldn’t you have a multiple winner bonus?

Also, a historically high jackpot might attract a historically high number of enthusiastic economists to play – though I don’t know how you would estimate that with any confidence.

104 Jim B. January 9, 2016 at 3:55 pm

Maybe a negative E(x), but a thick upper tail and almost no lower tail. Fourth moments matter!

105 dux.ie January 9, 2016 at 6:51 pm

Re previously: people who won lotteries regularly

http://arstechnica.com/tech-policy/2015/04/prosecutors-suspect-man-hacked-lottery-computers-to-score-winning-ticket/

“Prosecutors say they have evidence indicating the former head of computer security for a state lottery association tampered with lottery computers prior to him buying a ticket that won a \$14.3 million jackpot, according to a media report.”

https://en.wikipedia.org/wiki/Hot_Lotto_fraud_scandal#Subsequent_allegations

On November 23, 2005, Tipt*n’s brother Tommy won a \$568,990 jackpot prize share in a Colorado Lottery drawing. Edward Tipt*n was among those who constructed Colorado’s random number generator. A friend of Tommy Tipt*n claims that Tommy was asked to claim the ticket in return for 10 percent of the winnings.[8]
A December 29, 2007 prize of \$783,257 in a Wisconsin Lottery “Wisconsin’s Very Own Megabucks” drawing was paid to a limited liability corporation controlled by Robert Rhodes, who had also been charged in the 2010 Hot Lotto case. Records show that the corporation subsequently transferred money to Tipt*n.[8]
A November 23, 2011 Hot Lotto jackpot of \$1.2 million was paid to Kyle Conn, an owner of a construction company in Texas. Conn bought the ticket from the Oklahoma Lottery.[4]

106 Erhan Altay January 9, 2016 at 10:06 pm

The Federal tax rate on the Powerball jackpot is 25% that is much lower than your 40% estimate. As others stated, there are minor prizes to consider as well. As someone who considers himself economically literate, I’m in for \$100 this drawing!

http://taxfoundation.org/article/lottery-tax-rates-vary-greatly-state

107 WC Varones January 9, 2016 at 10:16 pm

Erhan,

The 25% is withholding, not the tax rate.

The tax rate is full ordinary income, plus Obamacare tax I believe: 39.6%.

108 Stephan January 10, 2016 at 2:01 am
109 Peter January 10, 2016 at 1:21 pm

The best I read about the lottery was where else can you spend a couple of bucks for a chance to tell your boss to go screw himself.

110 BrianMacker January 10, 2016 at 1:57 pm

Don’t forget that to get these kinds of payouts you have to buy exactly one ticket. The second you buy two tickets in the same lottery you have to divide the expected payout in half.

111 Rob Steele January 10, 2016 at 3:12 pm

Even if you only play on days with a positive expected value you have a negative expected value for playing over multiple days. The moral: play once if you must, then stop.

112 tim johnson January 10, 2016 at 5:46 pm

I have a much more simple concern: the way Powerball – and the media, which is worse – report that “jackpot” as if that is the actual worth of the winning prize. When it’s the “annuitized” value, counting the interest earnings given to the winner if the winner lets Powerball keep the money for 29 years while doling out yearly amounts.
Isn’t it pretty equivalent to saying your house is worth the sales price plus the interests payments on the 30-year mortgage? As in, inaccurate. No house is worth that, it’s worth the sales price, no?
And therefore, is it not baldly inaccurate to refer to the Powerball “jackpot” amount as the prize, the winnings, or any other term that would imply or state that the winner gets that much money at the point of winning (not speaking to the taxes owed on any such income, of course; an issue that is post this issue, just as we don’t refer to someone’s salary as not worth the gross just because it’s pre-tax. It’s still real money; just the “cash option” lump sum is the actual value of any Powerball win….no?.)

On the more esoteric issues raised above: I vaguely grasp what one friend calls the vig, or vigorish, or share of the player’s dollar that goes into the prize pot. He says since it’s so low in Powerball he doesn’t bother to play. I figurely, vaguely and can’t prove it, that the giant sizes of the Powerball pots make vig figuring irrelevant, as long as the player keeps the ticket purchase modest.
I like the idea above that playing such lotteries is a kind of “insurance” against not being in the pool of possible winners.

I also think much nonsense is said in comparing the odds of Powerball, 1 in 292 million, to such things as being struck by lightning or being elected President. Those seem wrongheaded: any happenstance that involves human behavior and choices and really acts of Nature, violates too much the need for pure randomness and such odds can’t be compared.
The fact that about 320 people in the United States are hit by lightning each year does not mean each American has a one in a million chance of being hit by lightning; same with Presidential hopes, etc. …no?

113 Winand von Petersdorff January 11, 2016 at 3:48 pm

Great post. You mentioned the 25 Percent chance that the Computer picks the same number twice. In your calculation that would bring the jackpot down to 372 Million( 496 Million *0.75= 372 Million). But is that right? If two winner share the numbers they share the price. But they still win. In my (probably wrong) calculation, the net Jackpot would be 0,75*496 Million + 0.5 * (496 Million – 372 Million) = 434 Million Dollar. Correct me if i am wrong

Winand

114 fritz lange January 11, 2016 at 6:56 pm

Ok with 1.2 billion payout possible with multiple winners also possible.i would like to imagine 1200 million prizes paid to 1200 individuals across country. This stimulus would be bigger then any bailout in history.

115 fritz lange January 11, 2016 at 7:09 pm