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If the odds of winning a lottery jackpot are one in 292 million, and the jackpot prize is expected to be greater than than $1 billion, does this make a lottery ticket a wise investment?

I know there is a risk that I'll lose my investment, and also a risk that if I do win then I may have to share the jackpot with other winners. However, there are also some other, smaller prizes available.

For example: if you were to take a dollar from me on the promise that you'd give me $3 back if i guessed the result of a single, fair coin toss, then I'd probably take this offer (as the odds of me guessing correctly are one in two, yet the potential returns are threefold).

Reference

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Chris Payne
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    I think this question is on topic because it's asking if playing the lottery can even have a positive expectation. That would make it an investment, not a gamble. I posted an answer to that effect. If showing people that the lottery is not a wise use of money isn't part of personal finance, what is? – Rocky Jan 10 '16 at 18:21
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    @Rocky In that case, I would suggest you edit the question to generalize it as being about lotteries in general, and not Powerball in particular, because then it becomes a question focused on the loss/expense avoidance aspect than about the odds or mechanics of a specific lottery ... but the OP designed this question to be specific to Powerball. Or perhaps pose such a question and re-post our answer there. e.g. "What impact does playing the lottery have on one's financial well-being?" – Chris W. Rea Jan 10 '16 at 19:01
  • I've posted on meta about this: http://meta.money.stackexchange.com/questions/2227/why-are-gambling-and-wagering-off-topic – GS - Apologise to Monica Jan 11 '16 at 06:33
  • Comments are not for extended discussion; the general chat about lotteries has been moved to chat and further comments may be deleted without being moved. – GS - Apologise to Monica Jan 12 '16 at 22:12
  • One point I haven't seen made in other answers is that the answer depends in part on the distribution of other ticket buyers numbers. The numbers you choose can not change your chance of winning (in a fair lottery), but they can change the chances of having to share a jackpot with other winners. Allegedly most players now use randomly generated picks, but in the early days of the UK lottery, you could get a slightly positive expectation on triple rollover weeks if you picked relatively unpopular numbers. – armb Jan 14 '16 at 15:36
  • You may find the most recent Planet Money podcast episode interesting - it covers exactly this. http://www.npr.org/sections/money/2016/01/13/462982983/10-11-51-52-62-18 – Oliver Cooper Jan 15 '16 at 02:41
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    Why do people always use the word "investment" when they talk about gambling in the lottery? Is the lottery your 401K? I play regularly both the lottery and blackjack, but I consider money I put down on either of those things as being gone to begin with. I might as well have set that money on fire. – L0j1k Jan 15 '16 at 21:59
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    The problem with the US is that you get taxed on lottery wins, making the winning somewhat like ordinary or investment income. In Australia we don't get taxed on gambling winnings (including lottery winnings), so gambling winnins is not viewed at all like an investment, instead it is viewed as it should be, as gambling. –  Jan 16 '16 at 21:31
  • If it was, the lottery would not exist. Simple as that. – Ben George Jan 18 '16 at 04:19
  • Remember too that most lottery winnings are taxable. So even if the expectation value is >1 for the lottery itself, the actual result may be much lower than that. (On the other hand, IF you have kept all the evidence, I believe you can deduct your lottery losses from winnings for tax purposes. Not from anything else, though.) – keshlam Dec 04 '23 at 23:53

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You're asking if lottery ticket can ever produce a positive expected value (EV). The short answer is, "no". There's an interesting article that goes into the details and is heavy on the math and graphs. The key point:

Even if you think you have a positive expected value due to the size of the jackpot being larger than the number of possible numbers, as more tickets are purchased (and the jackpot grows larger) the odds of someone else picking the winner goes up and your EV goes down. The article concludes:

[It] ... paints a grim picture for anyone still holding out hope that a lottery ticket can ever be an economically rational investment. As the jackpot grows in value, the number of people who try to win it grows super-linearly. This human behavior has a mathematical consequence: even though the jackpot itself can theoretically grow without bound, there is a point at which the consequent ticket-buying grows to such a fever pitch that the expected value of the jackpot actually starts going down again.

Rocky
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  • That article was an interesting read, and provided the data that proves that these tickets will probably never have a larger expected value than their purchase price. – Chris Payne Jan 10 '16 at 18:50
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    I think this answer is probably correct from an EV point of view. I would note, however, that EV is only meaningful in the limit of large numbers. If you're only going to play the lottery once, then EV does not tell you anything as an individual player with a small number of tickets on a single draw. –  Jan 10 '16 at 22:35
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    EV is EV. Your expected value per ticket is the same regardless of how many tickets you buy. Even with a single lottery ticket the odds of winning ANY prize is around 1:25 (using Powerball as an example). – Rocky Jan 10 '16 at 23:36
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    My point my have been too subtle for this format. If it catches anyone's eye later, it's probably better pursued in a math or a statistics forum. Indeed, "EV is EV" since that's a tautology, and, again, I don't think the answer is wrong insofar as it says it's talking about EV and is apparently accurate. On the other hand, it's common but logically flawed to confound a single draw from a distribution (e.g. buying one ticket one time) with either the average behavior of the distribution ("EV") or some other property of the distribution. Playing just an unusually large jackpot could fit this. –  Jan 11 '16 at 00:58
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    Many years ago, the Virginia lottery sometimes had positive expected values. An Australian consortium bought up as many tickets as possible. The consortium was lucky -- they won the lottery, despite not being able to buy all of the number combinations. – Jasper Jan 11 '16 at 03:51
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    It's almost as if people who are good with money and good with math have found a way to trick people who aren't good at money or good with math. Basic humanity in a nutshell, honestly. – corsiKa Jan 11 '16 at 06:08
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    @corsiKa, people who are good with money don't waste their money on playing lotteries as lotteries are for losers. –  Jan 11 '16 at 06:16
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    The Virginia thing wasn't a matter of luck; what made it a good play was that the subsidiary prizes got much bigger than they should have (people weren't buying tickets because they despaired of ever winning the jackpot, since it carried over so often, and the subsidiaries seemed too small to play for when set against the enormous jackpot). When they get big enough - something that only revenue-hungry politicians can force, since it's mathematically insane - it is actually possible to guarantee a total winnings at least equal to the jackpot with the right subset of combinations. – Stan Rogers Jan 11 '16 at 07:02
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    I particularly like the concluding quote in the article: "For the last word on this topic, however, I cede the floor to Durango Bill, who aptly observes that driving to the store to buy a Mega Millions ticket is more likely to be fatal than it is to make you rich." Though that's only tangential as it illustrates the low probability of winning, no the EV. – Lilienthal Jan 11 '16 at 12:04
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    @Lilienthal See this question on Skeptics: Are the chances of dying on the way to get lottery tickets larger than the chance of winning? – gerrit Jan 11 '16 at 14:21
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    "Can a lottery ever produce a positive EV" ? Yes - it did on Saturday in the UK - http://www.theguardian.com/science/2016/jan/09/national-lottery-lotto-drawing-odds-of-winning-maths - See Mawg's answer below. Yes - it requires extra rules, but those rules existed. – matt freake Jan 11 '16 at 16:24
  • A guy bought millions of tickets in a Florida lottery years back (and messed up and didn't cover the numbers he needed to). He won and won a lot of money. This article might have been written by him to strike fear in others. – blankip Jan 11 '16 at 22:43
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    You assume that your target is maximizing EV. This is not an obvious choice, and not always representative of motivation behind buying lottery tickets. I.e., if you want to maximize the probability of being worth one billion dollars within next month, then buying a lottery ticket is the wisest way to do it from the mathematical standpoint. – sashkello Jan 11 '16 at 23:23
  • One interesting thing about the article is that it predates the powerball changing the odds which reduces the odds of a tie. I don't doubt that the general conclusion is the same but it would change the numbers – Dean MacGregor Jan 12 '16 at 15:42
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    Conclusion is probably still right, but I'm not so sure that source article is accurate for this case. It seems to conflict some with this Slate analysis One issue is that the quoted article's NOT using powerball-specific odds or ticket volume history. But a bigger one may be this (also an issue in the Slate one): It's discounting the winnings for taxes, but not discounting the ticket price. And technically, they both should be, since gambling losses are deductible against winnings, at least in the US. – Jaydles Jan 12 '16 at 16:52
  • @sashkello: You would still have zero probability of being worth $1B next month by virtue of just the lottery, since the lump sum is less than $1B, you will still have to pay taxes, and the prize will be split. The current cash value is $930M, which is $562M after taxes, and will most likely be split by three, leaving you with a meager $187M. That hardly makes it even worth playing. – Mark Adler Jan 13 '16 at 04:43
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    @Jaydles: you can see my analysis on Quora which looks at the case where you buy all 292M possible tickets (which is the most efficient way to win), and where you deduct the $584M you spent on the tickets before taxes. – Mark Adler Jan 13 '16 at 04:48
  • @MarkAdler, good read, thanks. First I've seen that attacks the taxes on both sides! – Jaydles Jan 13 '16 at 14:40
  • I was confused to learn from the referenced article that lottery winning are taxed as income (in the US). Taking an honest view at lotteries where the sum of all payouts is less than the sum of all tickets bought, some have coined the difference between these two numbers a "math-inaptness-tax", so applying income tax would mean double taxation!? – Hagen von Eitzen Jan 13 '16 at 19:04
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    I downvoted this answer. While the linked article is probably correct in claiming that Powerball and Mega Millions never produce positive EV on buying a lottery ticket, it is incorrect to extend this claim to 'buying a lottery ticket never has positive EV'. 200_success's answer is far better, and indeed, has evidence that buying a lottery ticket has, in fact, been a positive-EV activity in the past. – senshin Jan 16 '16 at 17:13
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    @Jasper The consortium you mention was run by this guy. It was his 13th lottery win. https://lifethroughamathematicianseyes.wordpress.com/2013/09/08/stefan-mandel/ And he says it can't be done any more, there are too many numbers to print off the tickets in time. Here's Planet Money's interview with him. – Draco18s no longer trusts SE Jan 16 '16 at 17:44
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The other answers here do an excellent job of laying out the mathematics of the expected value. Here is a different take on the question of whether lottery tickets are a sensible investment.

I used to have the snobbish attitude that many mathematically literate people have towards lotteries: that they are "a tax on the mathematically illiterate", and so on. As I've gotten older I've realized that though, yes, it is certainly true that humans are staggeringly bad at estimating risks, that people actually are surprisingly rational when they spend their money. What then is the rational basis for buying lottery tickets, beyond the standard explanation of "it's cheap entertainment"?

Suppose you are a deeply poor person in America. Your substandard education prepared you for a job in manufacturing which no longer exists, you're working several minimum wage jobs just to keep food on the table, and you're one fall off a ladder from medical-expense-induced total financial disaster.

Now suppose you have things that you would like to spend truly enormous amounts of money on, like, say, sending your children to schools with ever-increasing tuitions, or a home in a safe neighbourhood.

Buying lottery tickets is a bad investment, sure. Name another legal investment strategy that has a million-dollar payout that is accessible to the poor in America. Even if you could invest 10% of your minimum-wage salary without missing the electricity bill, that's still not going to add up to a million bucks in your lifetime. Probably not even $100K.

When given a choice between no chance whatsoever at achieving your goals and a cheap chance that is literally a one-in-a-million chance at achieving your goals the rational choice is to take the bad investment option over no investment at all.

Eric Lippert
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    I agree with your analysis. Now, why do people pool their money to group buy lottery tickets? There probably is a rational reason, but I don't know it. – emory Jan 11 '16 at 16:25
  • There's a lot going on in this answer. +1 – elrobis Jan 11 '16 at 16:29
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    @emory: I don't know. I could guess. Two guesses: (1) the marginal utility argument. Consider a 1 in n chance of a 100 million bucks vs a 100 in n chance of a million bucks. The latter may have higher utility. Pools enable me to tweak the odds and payouts. (2) community effects. The same calculus that says lottery tickets are the only good investment vehicle for poor individuals holds for poor communities as well. – Eric Lippert Jan 11 '16 at 17:04
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    @EricLippert -- If your goal is to obtain 1 M$, and the prize is 800 M$, then you can maximize your chance of reaching your goal by pooling your money to get a small stake in a larger number of chances to win. – Jasper Jan 11 '16 at 17:10
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    @EricLippert -- There is an investment opportunity available that gives poor people the chance to retire with hundreds of thousands of dollars. But this investment requires waiting 40 or 50 years to reach that valuation. The opportunity: Cut your tobacco budget in half. Invest the savings in a Roth IRA in the stock of Philip Morris, with automatic dividend reinvestment. – Jasper Jan 11 '16 at 17:19
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    I completely disagree with this answer because the closing phrase, "no chance whatsoever at achieving your goals and a cheap chance that is literally a one-in-a-million chance at achieving your goals" calls the cost of buying the lottery tickets "cheap". But all the money spent on lottery tickets over a lifetime could have meant a life-saving medicine or medical treatment for a loved one! I object to the way you've framed it. Goals are always be pursued in the framework of a value system, but your answer pretends the value system doesn't exist, or is automatically slanted one way. – ErikE Jan 11 '16 at 21:02
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    Gamling is not an investment. –  Jan 11 '16 at 23:51
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    On the other hand, very bad things tend to happen when poor people come into sudden wealth. You could very well have to cut off your friends and family if you want to keep that money. Not to mention, it's hard to learn to spend or invest a lot of money, especially if you have never had experience managing money before, and have no idea whom to trust to help you. You could very well be better off not winning the lottery at all. – 200_success Jan 11 '16 at 23:56
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    Does stackexchange flag every question you answered as hot?-- Anyway. I very much like this argument; it's an argument I have made for the longest time. I'll elaborate on it more in an own answer. – Peter - Reinstate Monica Jan 12 '16 at 12:07
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    "is literally a one-in-a-million chance " actually it isn't 1:1M it is 1:292.2M. Further, the explanation you've given explains why people should view the lotto as a fun activity because it allows us to fantasize about receiving that much money. That it could result in a million dollar payout like nothing else would are the words of someone selling worthless junk to a naive person. – Dean MacGregor Jan 12 '16 at 15:46
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    People actually are surprisingly rational when they spend their money. [citation needed] – Zenadix Jan 12 '16 at 16:23
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    @ErikE: I'm not saying that this is good reasoning, or for that matter, a good idea. As you note, the perception that the cost is small may be inaccurate when integrated over time. Rather, I'm saying that we ought not to simply dismiss people who buy lottery tickets as mathematical ignoramuses or gambling addicts. – Eric Lippert Jan 12 '16 at 16:23
  • Eric, thanks for the clarification. I can see what you mean and will think on it. – ErikE Jan 12 '16 at 16:43
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    @ErikE: Ponder also the nature of the "poverty trap". There are programs designed to help the poor which, due to the way they phase out at certain income and wealth levels, produce a disincentive to increasing income and savings. If earning more, saving more and spending less makes you less well off in the short term, spending an extra dollar in your pocket on a lottery ticket seems pretty rational all of a sudden. – Eric Lippert Jan 12 '16 at 16:52
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    Although if you put that money in the bank you will have won...$100K! 100% guaranteed! Don't think there's a lottery in the world that will give you those odds. – Raydot Jan 13 '16 at 01:25
  • Had you continued the argument I thought you were going to make, that the enjoyment of playing the lottery is worth a dollar even if (or should I say when) you do lose, I may have been wiling to accept the argument. As posed I disagree. Studies show money doesn't buy happiness, but the poorer you are the closer it comes. That is to say 50 dollars to someone at poverty level is likely to be a far more significant boost to happiness then to the middle class. By this argument not only is the lottery a net loss of profit, but even winning it does far less for happiness the 5% of earnings – dsollen Jan 15 '16 at 16:08
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    @dsollen "Studies show money doesn't buy happiness" while that is one of those popular sayings, all the studies I've seen show a strong correlation between income (compared to your peers) and subjective well-being up to a certain income level (it also depends how you spend the money). See e.g. Economic Growth and Subjective Well-Being, Stevenson, Wolfers, 2008. Any current sources to that claim? – Voo Jan 16 '16 at 14:26
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    @Dave I have honestly no idea about the cost of lottery tickets or how people usually play, but wikipedia and the powerball site tell me prices are drawn twice per week and one ticket costs $2. So if we assume 2% interest pa and playing religiously we'll have... pretty much no money whatsoever after 40 years of playing. If you asked me if I'd prefer $15k USD in 40 years or a teeny tiny chance for a million.. ok I'd probably take the guaranteed payout, because I can already afford anything I really want - but if I couldn't and feared that my kids couldn't go to university? – Voo Jan 16 '16 at 14:37
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    @dsollen: Money can't buy everything, but did you ever try to buy anything without money? Money can't buy happiness but it can buy freedom from a great many unhappinesses. – Eric Lippert Jan 16 '16 at 14:37
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    @Voo but why make things worse? I was once at an NYC newspaper stand in line behind a guy who told me he spends $20 a day on the lottery. He did not look to me like he had a lot of money to spend, and that's $7,300 a year! I said "Wouldn't you be better off putting that money in the bank?" His reply "First I'll win and THEN I'll put it in the bank." Huh?! In 10 years that's $73K -- enough to put your kid through college. Guaranteed. So I guess you'd have to find your optimal price point where it's worth it to play regularly. – Raydot Jan 19 '16 at 15:14
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    @Dave I doubt that poor people have $600 to spend on lottery tickets per month. But if it's $20? Probably doable and you're not going to send anyone to college for ~$3,000. So it's a minimal chance vs none - seems reasonable (now if you have enough disposable income to avoid having to bet on luck? Yes, stupid idea we all agree. Not even a particularly interesting optimization problem if you have the numbers). – Voo Jan 19 '16 at 16:05
  • @Voo Yeah well, just an example. I hate to make assumptions but this guy didn't look too rich and the compulsion seemed a bit desperate which I take as a sign that he probably couldn't afford it. And also I'm not sure where you're getting $3000 from -- just counting one month? Putting $20 in the bank a day * 10 years is roughly $73K, not counting interest. As I mention earlier, no lottery in the world gives you a 100% chance to win that kind of money. Seems like an obvious choice to me. – Raydot Jan 19 '16 at 16:14
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    @Dave that's $20 month (as I said way more realistic for actually poor people) give or take over ten years. And no it's not an obvious choice. How much money would it be worth to you to get a k% chance to send your kid to college? That's a very realistic problem for many people and your "obvious choice" there implies that it'd be worth nothing (yes you end up with more money but not enough to send anyone to college). – Voo Jan 19 '16 at 16:34
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    I'll remind you all again that my point is neither (1) this is a good financial plan, nor (2) that everyone who plays the lottery does so from sound reasoning. My point is that we need not attribute only ignorance, stupidity, entertainment value or gambling addiction to people who play the lottery who seemingly cannot afford to do so. Incentives can be perverse, but they are still incentives. – Eric Lippert Jan 19 '16 at 18:02
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    EricL's point is obvious and correct. In my take: To put it extremely bluntly, there are historical periods when poor people are screwed. We are living in the ultimate example of such a historical period. There are other historical milieu where poor people are not screwed quite so much. Although it would be difficult to quantify, I'd bet that more people play the lottery in poor people are screwed eras. The common cry that lottery popularity is a "tax on stupidity" is wrong. Lottery popularity is an indication of astounding opportunity and wealth disparity. Such as today. – Fattie May 04 '16 at 15:37
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If you just buy a few lotto tickets normally, then no, it's not going to be a good investment, as @Jasper has shown.

However, there are certain scenarios where you can get a positive expected value from a lottery.

In 2012, it was revealed that some MIT students found a scheme to game the Massachusetts state lottery. The game, called Cash WinFall, had a quirk in the rules: the jackpot prize was capped at $2 million. Any money in the jackpot beyond $2 million would increase the payout of the consolation prizes. Thus, the game would sometimes have a positive expected value. The return on investment was 15% to 20% — enough for the participants to quit their jobs.

This specific loophole is no longer available: a cap was placed on the number of tickets sold per store, then the game was discontinued altogether.

Another possible strategy is to buy enough tickets to nearly assure a win, as one investment group did in 1992. Given a large enough jackpot, this strategy can yield a positive expected value, but not a guaranteed profit.

Caveats include:

  • You need to plunk down a lot of cash up front, and you will probably take the payout over many years.
  • The jackpot might get split among multiple winners. If multiple groups try this strategy, then they all lose. Also, the larger the jackpot, the higher the participation rate among the public, and the greater the chance that some random player will get lucky.
  • You need enough time to actually make the purchases. There is no shortcut where you can just say that you bought one of everything.
  • Lotteries may have rules to discourage bulk buying. For example, individual buyers may be given priority, which may slow down the bulk purchase enough to make it impractical.

Or, you might be a genius and exploit a flaw in the lottery's pseudorandom number generator, as one statistician did in an Ontario scratch-off lottery in 2011.

200_success
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    This should be the accepted answer. I was going to write this, there are definitely times when you can mathematically and statistically win money from the lottery. – enderland Jan 14 '16 at 14:15
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    @enderland I think those times are rare, and the OP is clearly asking about the Powerball given the specifics provided. Most lotteries do not ever have a positive EV. – Joe Jan 14 '16 at 15:54
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Others have already explained why lotteries have negative expected value, so in that sense it is never wise to buy a lottery ticket.

I will provide an alternative view, that it is not always unwise to buy a lottery ticket even though the expected value of the lottery ticket is lower than its cost (i.e. a loss). The question is what you mean with "wise"

A (not completely unlikely) scenario is one where your life (financially) suck, and even if you saved the cost of the ticket (instead of buying it) your life would still suck. Even if you saved the cost for a ticket every week for 10 years, your live would not be essentially better. You could maybe afford a TV, or a new car in 40 years, but if you were to quantify the happiness of your life it would still be essentially crappy. But winning the lottery would significantly improve your life and make you happy. So in this scenario there are two choices, either save the money for 0% chance of a happy life, or spend it on a ticket for a (extremely) small chance of a good life. Yes, the expected value of saving the money is higher than when buying the ticket, but "expected happiness" is higher when buying the ticket (non-zero).

This is clearly an extreme example, but variants of this might apply (the essence is that your valuation of the money is non-linear, 1 million will make you more than 1000 times as happy as 1000.)

epa095
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    Another example is where you have $1, and the mafia will break your kneecaps if you don't pay them $1m by the end of the week... – Ben Millwood Jan 11 '16 at 14:00
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    This is exactly why people who do understand the math still buy lottery tickets (I am one of them). Reducing lottery betting to a simple probability analysis is naive. Although the chance of winning is very low, the consequence of winning is extremely significant and is worth risking the negligible outlay. – Oscar Bravo Jan 11 '16 at 16:01
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    Ha! I like this, non-linearities in value assessment are so often over looked by myself and others. – Sam Jan 11 '16 at 16:34
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    In support of this argument, consider that the optimal number of tickets to purchase to balance -EV against the possibility of winning is one, as this increases the odds of winning an infinite amount from zero to some tiny but non-zero amount, for a minimum outlay (a buck or two). –  Jan 11 '16 at 17:33
  • that sounds like Pascal's wager. – stephenbayer Jan 13 '16 at 20:39
  • This sounds like justification for not financially planning if you aren't on a golden-train for financial success. – enderland Jan 14 '16 at 14:16
  • @Michael This reasoning shows you should buy one ticket, but when it is revealed that you have not won, you should buy another ticket. – Steven Gubkin Jan 16 '16 at 03:51
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The billion dollar jackpot is a sunk cost, a loss for prior bettors. If you had $292M and could buy every ticket combination, you'd be betting that not more than 2 other tickets will win on the next drawing. Even if 3 won, you'd have all the second place, third place, etc tickets, and would probably break even at worst.

Forget this extreme case. If I gave you a game where you had a chance to bet $100,000 for a 1 in 9 chance to win a million dollars, would you do it? Clearly, the odds are in your favor, right? But, for this kind of money, you'd probably pass.

There's a point where the market itself seems to reflect a set of probable outcomes and can be reduced to gambling. I've written about using options to do this very thing, yet, even in my writing, I call it gambling. I'm careful not to confuse the two (investing and gambling, that is.)

JTP - Apologise to Monica
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    Don't forget, depending on jurisdiction, the numbers get even worse when considering taxes and lump sum fees... You're realistically betting that not more than 2 other tickets win! – corsiKa Jan 11 '16 at 06:09
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    It would cost over $ 584 million to buy every possible ticket combination. (But the bulk buyer might be able to negotiate a kickback from the retailer(s), because the retailers receive a several percent of their lottery ticket sales as commissions.) – Jasper Jan 11 '16 at 06:34
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    The "usefulness" of $1,000,000 is less than 9 times the "usefulness" of $100,000. (Is that an actual economic concept, or is it something I just made up?) – user253751 Jan 11 '16 at 10:37
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    @immibis The term you're looking for is utility. And you make an excellent point; it's almost universally agreed that the marginal utility of money decreases, the more of it you have. So the utility you'd gain from winning $1m is very likely (much) lower than 9 times the utility you'd lose from paying $100k, so in that sense it's sensible not to play. – Andrzej Doyle Jan 11 '16 at 12:00
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    That said, if you had the option to play Joe's game many many times (i.e. both that he's willing to keep offering the chance, and you can keep paying $100k over and over again even after a run of bad luck), then the situation changes. The law of large numbers would kick in and make it exceptionally unlikely that you'd lose money overall. (Arguably though, if you had enough cash/credit that you could afford dozens or hundreds of entry fees, your marginal utility curve is probably already flat...) – Andrzej Doyle Jan 11 '16 at 12:04
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    For many, a $1M nest egg would be enough to retire on, especially if there were other funds besides the $100k bet. I understand how utility starts to fall beyond $20M jackpots, but just $1M? Not really seeing that argument. – JTP - Apologise to Monica Jan 11 '16 at 13:12
  • @corsiKa you get to deduct all your losing tickets from your winnings on your taxes. This helps mitigate the size of your losses greatly. Although it doesn't put you in positive expected value territory. – CQM Jan 11 '16 at 13:34
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    @immibis: Though the marginal utility -- that is, the utility added by each new "copy" of a thing -- does go down as you have more of it, money has some nice properties. First, money is fungible; the 100th dollar you get has exactly as much buying power as the 1st. Second, the marginal utility of money decreases far more slowly than other things. Having someone give you $100 is around 100x better than them giving you $1. Having someone hand you 100 free big macs is nowhere near 100x better than having one free big mac. – Eric Lippert Jan 11 '16 at 14:21
  • @JoeTaxpayer: When I was a teenager there was a couple who won second prize -- around $100K -- in the Lotto 6/49 in Ontario. They spent all of their winnings on lotto tickets and thereby increased their odds of winning the several-million-dollar jackpot to around 1 in 500. They did not win the jackpot. Some people are willing to spend a lot of money for long odds bets. – Eric Lippert Jan 11 '16 at 14:24
  • @JoeTaxpayer, looks like your AAPL bet more than paid off. http://www.google.com/finance?chdnp=1&chdd=1&chds=1&chdv=1&chvs=maximized&chdeh=0&chfdeh=0&chdet=1420232400000&chddm=197064&chls=IntervalBasedLine&q=NASDAQ:AAPL&ntsp=0&ei=Hr2TVpC2CJeSefmYudgO – user2023861 Jan 11 '16 at 14:36
  • @JoeTaxpayer Let me put it another way then... the "crappiness" of losing $100k is more than 9x the crappiness of not gaining $900k. Especially if you only have $105k or so. – user253751 Jan 11 '16 at 19:52
  • @immibis: With opportunity costs, there is no distinction between losing and "not gaining" any given sum of money. – Kevin Jan 12 '16 at 06:13
  • @Kevin That was not the point of my comment. "Losing" contrasts with "winning", not with "not gaining". – user253751 Jan 12 '16 at 06:15
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I estimated that the mean expected cash value of a $ 1.00 MegaMillions ticket in the July 5, 2016 drawing was about $ 1.23 = $ 0.18 consolation prizes + 258,890,850:1 chance of winning part of a cash jackpot that increased from about $ 289.6 million to about $ 313.3 million.

I estimated that the mean expected cash value of a $ 2.00 Powerball ticket in the January 13, 2016 drawing was about $ 1.65. I estimated this as follows:

1.                  Long-term mean prizes / ticket:   $ 1.00  
2.                  Mean consolation prizes / ticket: $ 0.32  
3.                  Estimated cash jackpot:           930   million dollars.  
4.                  Previous estimated cash jackpot:  558   million dollars.  
                    --------------------------------  ----------------------  
5. = (3) - (4).     Estimated pot increase            372   million dollars.  
6. = (1) - (2).     Estimated pot increase / ticket   $ 0.68.  
7. = (5) / (6).     Estimated tickets sold            547.1 million.  
8.                  Odds of winning jackpot:          292.2 million to one.  
                    --------------------------------  ----------------------  
9. = e^(-(7)/(8)).  Chance next ticket not shared      15.4 %  
10.= 1 - (9).       Chance next ticket shared:         84.6 %  
11.= (8) * (10).    # shared combinations:            247.3 million.  
12.= (7) / (11).    Mean splits already of ""           2.21  
13.= 1 + (12)       Mean splits of next ticket of ""    3.21  
14.= (9)+(10)/(13). Mean shares of next ticket         41.72 %  
15.= (3)*(14)/(8).  Mean jackpot pay next ticket      $ 1.328  
                    --------------------------------  -------  
16.= (2) + (15).    Expected value / ticket:          $ 1.648

17.= (9). Chance of another roll-over: 15.4 % . (about two-thirteenths).

This estimate does not take taxes into account. (There are ways to minimize the tax bill.) And of course, almost 96% of tickets win nothing.

Notes:

  1. According to the Connecticut Lottery's 2014 audited financial statements (in the "Schedule of Profit Margins by Game Type, Year Ended June 30, 2014"), slightly under 50% of its Powerball and MegaMillions ticket sales go to prize pools. This matched the January 2016 PowerPlay odds: When the jackpot was above 150 M$, $ 0.493 of each $ 1.00 PowerPlay add-on bet went toward incremental prizes.
  2. According to "Powerball - Prizes and Odds" on January 9, 2016, $ 0.32 of each $ 2.00 non-PowerPlay ticket went toward non-jackpot prizes.
  3. As advertised on the Powerball home page on January 12, 2016.
  4. As advertised on the Powerball home page on January 9, 2016.

.

  1. A quick sanity check is to compare this estimated number of tickets sold, versus the number of winning tickets from the previous drawing. As advertised on the Powerball home page on January 13, 2016, the January 9, 2016 draw awarded 18,315,365 consolation prizes. According to "Powerball - Prizes and Odds", "The overall odds of winning a prize are 1 in 24.87." 24.87 * 18,315,365 = about 455.5 million tickets sold in a 3 day period. The January 13 draw had 4 days of ticket sales.
    This value (of 455.4 million tickets) is a rough value, because it is mostly based on one number that was drawn. If human players avoided (or preferred) the number between 1 and 26 that happened to be drawn as the PowerBall, the estimate would be distorted.

.

  1. Each ticket purchase is coordinated with only a tiny fraction of the other ticket purchases. Thus, we can approximate the number combinations as being independently chosen. If the odds of winning the jackpot are n:1, and m tickets are sold, the odds that no ticket wins are (1 - 1/n)^m. e = the limit as n goes to infinity of (1 - 1/n)^-n. Thus, for huge values of n, (1 - 1/n)^m is about e^(-m/n).

Updated for July 5, 2016 MegaMillions draw.

Jasper
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    This assumes a uniform distribution of the numbers played. Prior to random draws, that clearly was not the case since people are biased. I do not know if the "quick pick" system is totally random or if a "quick pick" prevents picks like 123456. If so, there might be tickets with a higher/ positive EV. – StrongBad Jan 12 '16 at 01:41
  • @StrongBad -- Good point. The last time I checked, about three-quarters of American big jackpot lottery tickets were purchased using the "quick pick" system. – Jasper Jan 12 '16 at 02:12
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Question: Does a billion dollars make you 1,000 times more happy than a million dollars? Answer: It doesn't.

What counts is not the amount of money, but the subjective improvement that it makes to your life. And that improvement isn't linear, which is way the expected value of the inrease in your happiness / welfare / wellbeing is negative.

The picture changes if you consider that by buying a ticket you can tell yourself for one week "next week I might be a billionaire". What you actually pay for is not the expected value of the win, but one week of hope of becoming rich.

gnasher729
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    There are things that a person can do with a billion dollars, that they cannot do one-one thousandth of with a million dollars. – Jasper Jan 11 '16 at 03:54
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    What you are expressing is the idea that expected utility is a different concept from expected value, and most people have diminishing utility curves as a function of money. Because of that, the expected utility of a lottery ticket, ignoring its entertainment value, is virtually always negative. – Daniel Douglas Jan 11 '16 at 06:04
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    The issue of utility is interesting. A $100k earning couple would need to win close to $4M to net enough to even consider retiring. For each person utility drops at a different level. Beyond that $5M level, I'm not spending more or buying vacation homes, I'm doing charitable work. The utility of the 100th million is no less than the 6th, when it comes to the charities I support. Maybe that just proves your point. – JTP - Apologise to Monica Jan 12 '16 at 15:25
  • @Jasper the things someone can do with a billion dollars that they can not do with less are probably things they shouldn't be doing and, regardless, will not make they much happier (I suppose the exception is that some people might be very happy giving away that kind of money to certain charities or improving the lives of many, but there are other ways to get that feeling without the money) – Bill K Jan 15 '16 at 22:11
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I realize that most posters are US based, but the UK on Saturday had its biggest ever payout (a miserable £60m).

Because of the rules there, the estimated "value" of a £2 ticket was between £3 and £5.

http://www.theguardian.com/science/2016/jan/09/national-lottery-lotto-drawing-odds-of-winning-maths

JTP - Apologise to Monica
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Mawg says reinstate Monica
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    I too am in the UK, but the UK lottery limits the jackpot size (it had to be won this week), which is what prompted this question. I wondered if the UK lottery limited the size of the jackpot to avoid this situation, whereas the US lottery did not. – Chris Payne Jan 11 '16 at 09:54
  • So, your accepted answer, which states "no", contradicts this? :-) I guess you didn't win, then ... – Mawg says reinstate Monica Jan 11 '16 at 09:58
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    I accepted that answer because it provided statistical data that describes how many tickets are expected to be sold given the value of the jackpot, along with data that describes the chance of a collision (sharing your jackpot), which together could be used to prove that the expected value of a ticket would not increase above the purchase price. And I didn't win, but I also didn't lose either as I personally do not play the lottery. – Chris Payne Jan 11 '16 at 10:12
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    That's quiet all right. I wasn't quibbling. Personally, I take one ticket a week in whichever country I currently happen to be living in, since the cost is minimal. I try to have many/most of my picks over 31, so as not to share the pot with those who pick based on birthdays. I am still waiting, but not considering moving back to the states for the big one. – Mawg says reinstate Monica Jan 11 '16 at 10:31
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    As far as I can see, the UK Guardian article here disproves the accepted answers conjecture that a lottery can never have a postive EV. – matt freake Jan 11 '16 at 16:24
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    @Mawg Instead of sharing with the people that are picking birthday you share with the people avoiding the people picking birthdays. I am not sure thats a good choice. – Taemyr Jan 12 '16 at 15:28
  • What ration would you guess there of each? (otoh, clever thinking can get you into trouble, though. In Germany, the most frequently chosen numbers by far are 1, 2, 3, 4, 5, 6. – Mawg says reinstate Monica Jan 12 '16 at 17:52
  • This article confirms that, by the UK rules, a £2 ticket would, on average, return £4.50 http://www.theguardian.com/science/2016/jan/09/national-lottery-lotto-drawing-odds-of-winning-maths – Mawg says reinstate Monica Jan 13 '16 at 12:55
  • @mattfreake The OP specified the Powerball rules, and as such the accepted answer is correct. – Joe Jan 14 '16 at 15:56
  • I don' see any mention of Powerball in the question. I imagine that it has been editted. Just goes to show that the answer depends on exactly how you specify the the question :-) – Mawg says reinstate Monica Jan 14 '16 at 18:54
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I think playing certain kinds of lottery is as economically sound as buying certain kinds of insurance.

A lottery is an inverted insurance.

Let me elaborate.

We buy insurance for at least two reasons. The first one is clear: We pay a fee to protect ourselves from a risk which we don't want to (or cannot) bear. Although on average buying insurance is a loss, because we pay all the insurance's office buildings and employee's salaries, it still is a reasonable thing to do. (But it should also be clear that it is unreasonable to buy insurance for risks one could easily bear oneself.)

The second reason to buy insurance is that it puts us at ease. We don't have to be afraid of theft or of a mistake we make which would make us liable or of water damage to our house. In that sense we buy freedom of sorrow for a fee, even if the damage wouldn't in fact ruin us. That's totally legitimate.

Now I want to make the argument that buying a lottery ticket follows the same logic and is therefore not economically unreasonable at all.

While buying a lottery ticket is on average a loss, it provides us with a chance to obtain an amount of money we would normally never get. (Eric Lippert made this argument already.) The lottery fee buys us a small chance of something very valuable, much as the insurance frees us from a small risk of something very bad. If we don't buy the ticket, we may have 0% chance of becoming (extremely) rich. If we buy one, we clearly have a chance > 0%, which can be considered an improvement. (Imagine you'd have a 0.0000001% chance to save a loved one from certain death with a ticket. You'd bite.)

Even the second argument, that an insurance puts us at ease, can be mirrored for lotteries. The chance to win something may provide entertainment in our otherwise dull everyday life.

Considering that playing the lottery only makes sense for the chance to obtain more money than otherwise possible, one should avoid lotteries which have lots of smaller prizes because we are not really interested in those. (It would be more economical to save the money for smaller amounts.) We ideally only want lotteries which lean on the big money prizes.

Peter - Reinstate Monica
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  • I'd love to hear a counter-argument. – Peter - Reinstate Monica Jan 13 '16 at 09:39
  • I make a similar argument in my answer: An event with a low probability can have a huge impact. I just wouldn't call it economically sound or reasonable, per se. I assume people didn't like your comparison with insurance as you can insure yourself also against higher risks with smaller impacts. I'd still agree that there are certain types of insurances or insurance-like products that have similar characteristics in terms of risk and payout. – vic Jan 13 '16 at 13:26
  • @vic and as I have pointed out, only those insurances (which insure against catastrophic events) are economically sound, despite their on average negative equity. – Peter - Reinstate Monica Jan 13 '16 at 14:28
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Firstly, playing the lottery is not investing it is gambling. The odds in gambling are always against you and with the house.

Secondly, no one would ever give you a payout of 3 to 1 when the odds are 50:50, unless they were looking to give away money. Even when you place your chips on either red or black on a roulette table your payout if you are correct is 100% (double your money), however the odds of winning are less than 50%, there are 18 reds, 18 blacks and 2 greens (0 and 00). Even if you place your chips on one single number, your payout will be 35:1 but your odds of winning are 1:38. The odds are always with the house.

If you want to play the lotto, use some money you don't need and expect to lose, have some fun and enjoy yourself if you get any small winnings. Gambling should be looked at as a source of entertainment not a source of investing. If you take gambling more serious than this then you might have a problem.

  • Then what would you call Blackjack? Is it gambling? I venture that it is not as you can learn to play BJ well enough to have an edge and make money in the long term. – KevinDTimm Jan 11 '16 at 16:28
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    I would say that games like BJ or Poker, while still being part of gambling (that is the face to face modes not the online modes or slot machine modes), the player does have some small control by the decisions they make, (I think poker more so than BJ purely because of the bluffing factor) and it may be possible for a limited few to gain a small edge, but other forms of gambling like roulette, slot machines, online and lotteries are purely games of chance where the odds are always with the house, as George mentions. – Victor Jan 11 '16 at 22:06
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    "Investing" is just legalized gambling, as you have no guarantees of a return in either case. In the case of investing, the house edge goes to the brokerage firms. Your mileage will not vary. – CodeGnome Jan 12 '16 at 23:00
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    @CodeGnome, just because you don't know how to invest does not mean it is gambling. In your definition even owning a business is gambling. Just because there is a risk does not mean it is gambling. Most people who do succeed in business and investing do so because they manage their risks. You cannot manage any risk with gambling, because the odds are designed to be against you and with the house. –  Jan 13 '16 at 02:51
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Gambling is never a wise investment. Even assuming that the stated odds are correct, there can be multiple winners, and the jackpot is shared between the winners, so the individual payout can be significantly less than the total jackpot. If I were to take a dollar from you and a dollar from your buddy on the promise that I'd give the two of you a total of $3 back if you both guessed the result of a single, fair coin toss, would you take the offer?

Note, also, that the "jackpot" value is quite misleading: it's the sum of the annual payments, and if you reduce that to present value it's significantly less.

Pete Becker
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  • True. But in the instance above, the jackpot is predicted to be over 4 times the odds of winning. That'd mean that I'd have to share it with 4 other people in order for the amount won to be lower than the odds of winning (granted I haven't taken tax and annuity payments into account). – Chris Payne Jan 10 '16 at 16:39
  • How many tickets are outstanding? – Pete Becker Jan 10 '16 at 16:41
  • I'm guessing you're implying that I don't know how many tickets are sold, and that there is no limit on the number of tickets that can be sold, both of which are true. I know it's a risk and by no means a dead cert, but aren't all volatile investments risky? – Chris Payne Jan 10 '16 at 16:47
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    If the odds of getting the winning number are 1 in 1000 and there are 10,000 tickets outstanding, chances are pretty good that each winner will share with about 9 others. Don't forget, the purpose of a lottery is to make money for the operator. If there was an easy way to beat it, they wouldn't run it. So any analysis that leads to the conclusion that you can beat the odds is probably wrong. – Pete Becker Jan 10 '16 at 16:49
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    @PeteBecker The operator in this case is safe - the jackpot is made up of old, already-lost lottery purchases that roll over every time no one wins it. You can't beat the odds playing every day, but you can beat the odds if you only play when the jackpot has gotten to insanely high numbers. – ceejayoz Jan 11 '16 at 05:29
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You can have a positive expected return on a lottery ticket purchase, but only if the lottery requires all players to pick their own numbers and doesn't have an option to buy a ticket with a randomly generated set of numbers.

This is because people are very bad at picking random numbers, and will tend to pick numbers that are fairly evenly spaced or based on dates rather than genuinely random numbers. For example in January 1995 the UK national lottery happened to have fairly well-spaced numbers (7, 17, 23, 32, 38 & 42), and there were 133 winners with all six numbers.

So they way to win is to wait for a draw where a rollover jackpot is high enough that your expected winnings are positive if you are the only winner, and pick a set of numbers that looks stupidly non-random, but is not so very non-random that people will have picked it anyway, like 1, 2, 3, 4, 5, 6. For a "pick 6 in the range from 1-49" lottery you might pick something like 3, 42, 43, 44, 48, 49. But it doesn't work if there's a random option, since a significant number of players will use it and get genuinely random numbers, and so your chances of being the only winner get much smaller.

Mike Scott
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  • I agree with your assertion that most people are less inclined to manually pick certain ranges of numbers. However, I disagree with your first sentence. Either it is mathematically possible to have a positive EV, or it is not possible. In either case, the existence of randomly generated picks does not change that answer. – TTT May 04 '16 at 15:23
  • @TTT The existence of randomly generated picks changes the likelihood of having multiple winners. If they only sold one ticket per combination, the expected value of the jackpot would be greater than the ticket price whenever the cash value of the lottery was more than the ticket price times the odds. But that's not the case. Shared wins reduce the effective jackpot value. That is discussed in more detail in the currently top-voted and accepted answer. And even more detail in the linked article. – Brythan Jul 04 '16 at 04:45
  • @Brythan - the conclusion of the linked article is incorrect, and the write admits he could be wrong at the end. It's easy to prove too: if the jackpot was $100 billion, of course you have a positive EV because not enough tickets will be sold for a single drawing so the ties will taper off. Whatever it is, the number is probably impractically high that we may never see it, but it is certainly possible. – TTT Jul 04 '16 at 13:07
  • Also, the main issue I have with the first sentence is the use of the word only. I'll admit that if people picked their own numbers there would be a skewed distribution, and the EV could shift as a result, but I don't believe manually picks is a requirement for positive EV. (Plus even if there was only manual, many people could think this way, or even use their own random generator tools, so the skew would be lessened a bit.) – TTT Jul 04 '16 at 13:19
  • How would the jackpot get to $100 billion (cash value)? That means that $400 billion in tickets were sold (the jackpot's cash value is roughly 25% of ticket sales since the last win). Since both lotteries have less than three hundred million combinations, that would mean sales of a thousand tickets for every possible combination. There'd have been a lottery with all combinations picked long before the jackpot got that high. – Brythan Jul 04 '16 at 18:50
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Lottery tickets where I live are often for charity. The charity does good things with your money. So you can buy a ticket and feel good whether you win or not, so that makes it an investment in your own well-being.

For some of us, who maybe buy a lottery ticket once a year, it's the fun you are paying for. You know you are not really going to win, but you spend a few hours being excited waiting for the draw. Cheaper than the cinema.

And you never know, you might win after all... The odds may be ridiculous, but somebody's going to get it...

RedSonja
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    While that's true, I can assure you that it would be easier to simply donate said money to charity. Not only will they get a larger amount that way (overhead to run the lottery) but it'll save them the bureaucratic hassle to apply for the money too. – Ghanima May 03 '16 at 19:50
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Possibly, if you can get them at a discount. But not if you have to pay full price.

Say there's a $1 million Jackpot for $1 tickets. The seller might sell 1.25 million of these tickets, to raise $1.25 million pay a winner $1 million, and keep $250,000. In this example, the so-called "expected value" of your $1 ticket is $1 million/1.25 million tickets= 80 cents, which is less than $1. If someone were willing to "dump" his ticket for say, 50 cents, what you paid would be less than the expected value, and over enough "trials," you would make a profit.

Warren Buffett used to say that he would never buy a lottery ticket, but would not refuse one given to him free. That's the ultimate "discount."

Larger Jackpots would work on the same principle; you would lose money "on average" for buying a ticket. So it's not the size of the Jackpot but the size of the discount that determines whether or not it is worthwhile to buy a lottery ticket.

Tom Au
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    The Jackpot usually comes from money not payed out in previous rounds. Ie. it's allowed to accumulate. – Taemyr Jan 12 '16 at 15:37
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A lot of these answers are really weak.

The expected value is pretty much the answer. You have to also though, especially as many many millions of tickets are purchased--make part of the valuation the odds of the jackpot being split x ways.

So about 1 in 290--> the jackpot needs to be a take-home pot of $580 million for the $2 ticket. Assume the average # of winners is about 1.5 so half the time you're going to split the pot, bringing the valuation needed for the same jackpot to be $870 million.

It's actually somewhat not common to have split jackpots because the odds are very bad + many people pick 'favourite numbers'.

George
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    If you made the assumption that a lot of people played numbers that made up dates that were significant to them, you could also reduce the chance of sharing your jackpot by only playing the numbers 32 and up. – Chris Payne Jan 11 '16 at 21:51
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Is playing the lottery a wise investment? --Probably not.

Is playing the lottery an investment at all? --Probably not though I'll make a remark on that further below.

Does it make any sense to play the lottery in order to improve your total asset allocation? --If you follow the theory of the Black Swan, it actually might.

Let me elaborate. The Black Swan theory says that events that we consider extremely improbable can have an extreme impact. So extreme, in fact, that its value would massively outweigh the combined value of all impacts of all probable events together. In statistical terms, we are speaking about events on the outer limits of the common probablity distribution, so called outliers that have a high impact.

Example: If you invest $2000 on the stock market today, stay invested for 20 years, and reinvest all earnings, it is probable within a 66% confidence interval that you will have an 8 % expected return (ER) per year on average, giving you a total of roughly $9300. That's very much simplified, of course, the actual number can be very different depending on the deviations from the ER and when they happen. Now let's take the same $2000 and buy weekly lottery tickets for 20 years. For the sake of simplicity I will forgo an NPV calculation and assume one ticket costs roughly $2. If you should win, which would be an entirely improbable event, your winnings would by far outweigh your ER from investing the same amount.

When making models that should be mathematically solvable, these outliers are usually not taken into consideration. Standard portfolio management (PM) theory is only working within so called confidence intervals up to 99% - everything else just wouldn't be practical. In other words, if there is not at least a 1% probability a certain outcome will happen, we'll ignore it. In practice, most analysts take even smaller confidence intervals, so they ignore even more.

That's the reason, though, why no object that would fall within the realms of this outer limit is an investment in terms of the PM theory. Or at least not a recommendable one.

Having said all that, it still might improve your position if you add a lottery ticket to the mix. The Black Swan theory specifically does not only apply to the risk side of things, but also on the chance side. So, while standard PM theory would not consider the lottery ticket an investment, thus not accept it into the asset allocation, the Black Swan theory would appreciate the fact that there is minimal chance of huge success.

Still, in terms of valuation, it follows the PM theory. The lottery ticket, while it could be part of some "investment balance sheet", would have to be written off to 0 immediately and no expected value would be attached to it. Consequently, such an investment or gamble only makes sense if your other, safe investments give you so much income that you can easily afford it really without having to give up anything else in your life. In other words, you have to consider it money thrown out of the window.

So, while from a psychological perspective it makes sense that especially poorer people will buy a lottery ticket, as Eric very well explained, it is actually the wealthier who should consider doing so. If anyone. :)

Community
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vic
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Here's an interesting link to a discussion about an Australian investor group back in the 1990s that bought almost every combination in the West Virginia lottery. It's pretty fascinating stuff.

How An Australian Group Cornered A Lottery

I don't need to add to what's already been said here, but it's a fun story!

Daniel Anderson
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Mathematically speaking there would be a point where the expected value EV of purchasing every possible ticket would be favorable but only if you take in account both the jackpot payout and the lesser payouts of all the wining tickets however practically speaking since the powerball has a liability payout limit which means they dont have to pay out more money than they took in you cant beat the house ( or the government)

revenant
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Lotteries are like the inverse of insurance policies. Instead of paying money to mitigate the impact of an unlikely event which is extremely negative, you are paying money to obtain a chance of experiencing an unlikely event which is extremely positive.

One thing to keep in mind regarding lotteries is the diminishing marginal utility of money. If you know you'll never use more than say $100 million in your entire life, no matter how much money you might acquire, then buying tickets for lotteries where the grand prize is over $100 million stops being increasingly "worth the price of entry".

Personally, I'd rather play a lottery where the grand prize is sub-100 million, and where there are no prizes which are sub-1 million, because I do not believe that any other amounts of winnings are going to be life-changing for me in a way that I am likely to fully appreciate.

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According to a financial adviser I spoke to, lottery is the riskiest of investments, whereas cash is the safest. Everything else falls between these 2 extremes.

JayJay
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    Marco, your financial adviser is a dill, because lotteries are not investments they are gambling. And holding cash is not an investment either (nor is cash in the bank), as by hold cash (or having cash in the bank) you are guaranteed to lose money over time due to inflation. I hope you didn't pay him for his services. – Victor Jan 11 '16 at 22:13
  • My bad, I didn't explain the context of the conversation. The financial adviser was providing some training on financial products to internal staff. He started the conversation, most likely to grab our attention with a question like 'What's the most risky way to invest your money? Hint: everyday people put money into it". He didn't suggest to 'invest' in the lottery and as far as I know he's quite successful. – JayJay Jan 25 '16 at 17:42