Cost justified lotto ticket?

October 13, 2005 10:52 PM

Many lotteries (see Powerball) increase the size of their potential jackpot over time. Assuming a $1 ticket cost, is it cost-justified to buy one ticket for every drawing when the value of the payout is greater than the odds against winning?

The PowerBall odds state that the odds of winning the jackpot are 1 in 146,107,962. Assume you only play when the cash value of the jackpot is greater than $146,107,962, and you only buy a single ticket for each drawing. Averaging out for the long run (though admittedly any individual will only be able to participate in a very small number of actual drawings), wouldn't your expected losses be less than the expected gains from winning the jackpot(s), making the purchase of a single ticket cost-justified?

Econ experts, please tell me why I'm a moron who deserves to pay the "stupidity tax."

posted by stopgap to work & money (27 comments total)

I wrote an article about this problem in Slate. (Hope self-link is OK if it answers the question, or tries to.)

(Summary: you might well not be a moron, and, if you are, it's probably not an economist or a mathematician who has the right to tell you so.)
posted by escabeche at 10:56 PM on October 13, 2005

This makes sense as long as you can be sure no-one else picks the same numbers as you.
posted by pompomtom at 11:13 PM on October 13, 2005

You're looking at (according to your profile) 43.98% tax load (35% federal, 8.98% state). As such, you'll need $260m to take home $146m.

Beyond that, powerball only pays the full amount if taken as a graduated annuity. Since this is a pretty raw deal for a savvy investor, you'll want to take the lump sum. This kills your take home by approximately 50%, so now the jackpot has to be about $520m to break even.

That is all discounted by the smaller payouts, which are worth about 20 cents on the dollar, according to the page you listed. This brings the required jackpot down to $416m.

After this, the only remaining issue is how many tickets are sold per draw, to factor in the chance of splitting the jackpot. Given the lack of easily googleable ticket sales statistucs, I can't make an accurate estimate, so I'll use a conservative factor of a 10% chance of splitting the jackpot.

Which would make powerball an even money game when the jackpot hits $416,000,000.
posted by I Love Tacos at 12:34 AM on October 14, 2005

I don't know how to paste this better... but here is a table I made in excel, which shows the value of powerball at $146m.

Advertised	Actual Pre-Tax	Actual Post-Tax	Odds	Inverse Odds	Value

 $146,107,962.00 	 $73,000,000.00 	 $40,150,000.00 	146107962	6.84E-09	 $0.275 

 $200,000.00 	 $200,000.00 	 $110,000.00 	3563609	2.81E-07	 $0.031 

 $10,000.00 	 $10,000.00 	 $6,000.00 	584432	1.71E-06	 $0.010 

 $100.00 	 $100.00 	 $100.00 	14254	7.02E-05	 $0.007 

 $100.00 	 $100.00 	 $100.00 	11927	8.38E-05	 $0.008 

 $7.00 	 $7.00 	 $7.00 	291	3.44E-03	 $0.024 

 $7.00 	 $7.00 	 $7.00 	745	1.34E-03	 $0.009 

 $4.00 	 $4.00 	 $4.00 	127	7.87E-03	 $0.031 

 $3.00 	 $3.00 	 $3.00 	67	1.49E-02	 $0.045 

					 $0.441 

The columns are the advertised payout, the actual payout, how much you'll get post-tax, the odds of receiving that, the odds in decimal form, and then the value that that prize provides, from a $1 ticket.

On the last line is the expected take-home for every dollar put into that $146m jackpot, excluding the chance of splitting the jackpot. 44.1 cents.

The lottery is a cruel joke.
posted by I Love Tacos at 12:52 AM on October 14, 2005

Doesn't it depend on why you're buying a ticket? Perhaps the fun of being able to daydream about winning is worth the buck a week.

The expected take-home for every dollar put into buying a movie ticket is $0, but that doesn't make movies a cruel joke.
posted by Justinian at 12:59 AM on October 14, 2005

I wrote an article about this problem in Slate.

I'm going to trust these numbers, namely the 37% expected share of the winner's jackpot. If you take this into account, then you have to let the jackpot get alot higher than the odds of 1 in 146,107,962. But then there's all those chances to win the smaller prizes too. It looks like once the jackpot gets over $317,081,136 then it makes sense to play.

On preview: Oh yeah, taxes. Since you're only going to take home 56.02%, that bumps it up to $566,014,166.
posted by 23skidoo at 1:00 AM on October 14, 2005

Well, movies don't promise to make you rich.

The odds of winning the lottery are not changed appreciably by buying a ticket. Just don't bother.
posted by kindall at 1:01 AM on October 14, 2005

Just realized, in my first post, the final number should have been $462,000,000 as powerball's breakeven point.

I take no issue with gambling or dreaming, but people should understand the odds before they play.
posted by I Love Tacos at 1:07 AM on October 14, 2005

Well, movies don't promise to make you rich.

Neither do lotteries, they offer the hope of a chance to get rich. That hope can be seen as having a value, as Justinian was suggesting.

The payback on lotteries varies with different lotteries in different territories. For example, the UK Lotto gives out 50% of the money collected in prizes. With enough rollovers then it is possible that the expected payout could exceed the total value of tickets bought for a particular round. However, since most payouts would be made across the range of bets in any given week it is only the continuing rollover value of the main prize that would really impact. Taking the latest game as typical, about 25% of the total prizepot goes to the jackpot, or about 12.5% of the total take. So assuming (for ease not reality) a stable income on tickets then 4 rollovers might see a prizepot equivalent to the total stake for the 5th round (4x12.5% + 50% for that week). UK Lotto rules give you a lump sum in all cases.
posted by biffa at 1:34 AM on October 14, 2005

pompomtom is right, the main thing you've missed is that an unknown number of other entrants may have the same numbers, in which case you'd have to share the jackpot with them.

Some people actually did this in Australia a couple of years ago.

They waited until the jackpot was greater than the cost of covering every possible combination, crossed their fingers that nobody had the same numbers, and made money. Not huge amounts, but the members of the syndicate doubled their money, maybe? I'll try and find a link.
posted by AmbroseChapel at 1:57 AM on October 14, 2005

I didn't take that into account but in terms of the theoretical payout I don't need to, in theory with the situation I describe then the total pot equals the total stake, that is, the return is 1:1. (This addresses the original question)
posted by biffa at 2:45 AM on October 14, 2005

I have completely revised my Powerball playing strategy. Now, instead of buying a ticket only when the jackpot hits $240 million, I'll just throw a dollar bill into the middle of a crowd. Far more entertaining.
The Lottery: You can't lose if you don't play!
posted by Floydd at 6:20 AM on October 14, 2005

I question whether the expected value even matters, in the real world where you buy 1 to 20 tickets. Your odds of winning are so low that it's a pure gamble no matter how much they are paying out (up to a point).

OTOH, the lottery is a very cheap way to get a chance to win a vast fortune. Nothing else comes close.
posted by smackfu at 6:36 AM on October 14, 2005

Think of a childhood friend you haven't seen in ten years. Pick up the phone. Dial an arbitrary number. If that friend answers, you just won the lottery.
posted by StickyCarpet at 6:42 AM on October 14, 2005

One other logical fallacy no one has pointed out: the odds of winning each drawing is 1 in 146,107,962. Buying one ticket in each drawing over a lifetime in no way increases your odds of winning any particular drawing. You'd need to buy at least 146,107,962 tickets in each drawing.
Imagine flipping a coin. On your first flip, your odds of getting heads are 1 in 2. On your next flip, your odds of getting tails are 1 in 2. Just because you flipped the coin once and got heads doesn't mean that the chances of it coming up tails on the next flip improve.
Basically, your investment doesn't make sense because you are assuming from the outset that, by buying one ticket for each drawing, you would eventually pick the winning numbers. In fact, nothing of the sort is the case.
posted by robhuddles at 8:22 AM on October 14, 2005

Most of the other commenters have covered the important things--

One, taxes make the necessary payout much higher (although in the u.s. your gambling losses are deductible to the extent of your gambling gains, which may help you a trivial bit).

Two, the serious risk is having to share the prize. As a few others have alluded to, however, not all numbers are created equal. Each number, of course, has an equal likelihood of winning, but because many people pick numbers non-randomly, you will profit by trying to pick the most unpopular numbers you can. There are a few papers and surveys that discuss tactics for this (people like numbers between 1 and 31, since they can represent lucky dates; people don't like numbers that look wildly unlikely to win like 37-38-39-41-40, etc.) If you pick unpopular numbers you minimize your probability of having to share the jackpot, and this helps.

Three, you are obviously still very very unlikely to win, even if the expected value of your purchase is positive.
posted by willbaude at 8:29 AM on October 14, 2005

To contrast with willbaude's statement a little, apparently the numbers 1-2-3-4-5-6 were very popular in the UK Lotto (and probably still are). I imagine it's people who heard that those were just as likely to win as any other six and thought they'd be clever. They are as likely to win but may be disappointed with what they get. From the article i saw that mentioned this I recall there are a few other common patterns, multiples of 6 is one, another being the 4 corner numbers on the ticket & any 2 others.
I seem to remember there was one where the 6 numbers were all under 32, resulting in a large amount of jackpot winners, each getting less than £100,000 (IIRC) against a normal (saturday) prize of multiple millions.
posted by biffa at 8:40 AM on October 14, 2005

When calculating the expected outcome, don't forget that there are plenty of non-jackpot winnings. So figure in your chance of winning $10 or a free ticket or $100, $500 etc.

The australian group already mentioned above obviously won all of those, too. I remember seeing a tv special on their strategy and how they had all the tickets catalogued and had to find *all* the winning tickets and get them in by whatever the deadline was.

They actually didn't end up buying every possibly combination because they ran out of time before the buying deadline. It's not like they could just plunk down a few million dollars and tell the lottery commision to put them in for one of each combination. They actually had to list all the combinations, fill out all the little sheets and take them in batches to convenience stores and such to buy the tickets (and of course convenience store owners were none too thrilled and that limited the batch size). They did manage to get all the tickets bought before the deadline, but I think they did win the jackpot and they won tonnes of smaller prizes as well.
posted by duck at 8:44 AM on October 14, 2005

Biffa:

Agreed with respect to 1-2-3-4-5-6. The sequence I mentioned is less popular both because of its weird starting point, and because it is slightly out of order. The number of people who buy 1-3-2-4-5-6 is much smaller.
posted by willbaude at 9:14 AM on October 14, 2005

The number of people who buy 1-3-2-4-5-6 is much smaller.

But that's equivalent to 1-2-3-4-5-6!
posted by kindall at 9:35 AM on October 14, 2005

You aren't a moron.

If I'm reading your strategy right your expected payoff is greater than the ticket price. But expectation value isn't really the right way to look at this problem because at your scale (buying one ticket every so often) the assumption of a linear utility function breaks down.

The ninth-grade statistics "proof" that everybody who buys a lottery ticket is too dumb to live always irritates me because it makes this stupid linearity assumption and fails to take into account that there's a threshold below which an amount of money lost doesn't matter.

Even if the expectation value is LESS less than the ticket cost, the utility difference between the jackpot and the ticket price might mean you should buy a ticket anyway. Say you buy 10 tickets a year. How does ten bucks out of your pocket over the course of a year affect YOUR life -- probably not at all. Having an extra hundred or so million, though, would be a big boost for anybody I know. Your psychology, financial situation and other personal factors determine the shape of your utility curve and whether you should get in (and how in you should get) for a given jackpot.

Also, as Justinian said, there are benefits that aren't taken into account in the classic analysis -- it is possible to have fun at the track even if all of your horses lose.
posted by Opposite George at 9:47 AM on October 14, 2005

Justinian said it best. I buy lottery tickets so that I have something to think about as I go to sleep, not with an expectation of winning.

The lottery here for Saturday is $22million (tax-free because it's Canada and we love our lottery winners), and odds are 1:~14,000,000. I think I'll go buy a ticket today with the strong belief that I will not win and the ticket will be better for the than the chocolate bar I could have spent the money on.
posted by Kickstart70 at 9:49 AM on October 14, 2005

Sorry. I'm unfamiliar with the particular lotteries at hand. In some cases order matters. If not, you want something like 1-3-4-5-6-7, but 35-37-38-39-40-41 will do even better.
posted by willbaude at 11:07 AM on October 14, 2005

Opposite George: The notion of utility is relevant but I think it suggests a far different conclusion than the one that you're drawing. You're still comparing the absolute value of $1 to the absolute value of $1,000,000. The point is to compare the relative value of your 20,000th dollar to your 150,000,000th dollar. This misanalysis is exactly why so many people misallocate significant parts of their budget to the lottery.

Whether we're analyzing the case of a wealthy person or a poor person, considering the utility function actually decreases the economic rationale for playing the lottery. However, when you consider the typical actual lottery player I think that the difference is far more significant. The typical actual lottery player is more likely to have a lower income and at the same time more likely to be gambling with more money.

Comparing the lottery to a night at the movies is only reasonable when you're considering a person who can afford a night at the movies.
posted by stuart_s at 11:37 AM on October 14, 2005

Like with all gambling, you should only risk money you won't miss. When what you're betting has the personal value of 0 to you, you can only win. Gambling isn't an investment.

Consider this base case. You meet up with an eccentric dental floss tycoon who wants to make you an offer. This is a one-time chance, a true once in a lifetime opportunity. You can pick one of two things, and after you make this choice he will kick you out of the mansion and never see you again. The choices are:

a) He gives you a penny.

b) You flip a coin. If it lands on the edge he gives you a million dollars. Otherwise, you get nothing. He informs you the chance that the coin will land on its edge is exactly one in 146 million.

You quickly calculate that (b) is a losing proposition on average, and so you walk out with your shiny penny. Did you really make the smart choice? Would you take (b) if the payoff were $146m, but not if it were $145m?

If you can only do it once then I think (b) is the better choice. If you're an immortal robot playing this game against the tycoon over and over again, (b) is a rotten choice.

But face it, you're not going to play enough powerball in your lifetime to see anything like statistical averages. It's not really about whether the jackpot is over $146m, it's about how much money you're willing to throw away on the lottery over the course of your life. Whether you throw away a "stupid" amount depends on you.

Post-Jrun: I see Opposite George makes this point as well
posted by fleacircus at 2:13 PM on October 14, 2005

And, in my example, convert dollar amounts to penny amounts for analysis. Oops.
posted by fleacircus at 2:59 PM on October 14, 2005

Oh man, fleacircus, if I were an immortal robot, that'd be sweet!

Personally, I look at it from the standpoint that the dollar I spend on a ticket prevents me from buying that soda that I don't need at work. Not unlike Kickstart70, I guess.

Then again, at work, they have a lotto pool. They try to get me to buy in by saying how much it would suck to be the only one in the department not to win. To me, it just means that I'd know 30+ millionaires to mooch off of, and it wouldn't cost me anything!
posted by sysinfo at 9:56 PM on October 14, 2005