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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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    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, 2016 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, 2016 at 19:01
  • I've posted on meta about this: meta.money.stackexchange.com/questions/2227/…
    – GS - Apologise to Monica ♦
    Jan 11, 2016 at 6:33
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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, 2016 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.
    – user9722
    Jan 16, 2016 at 21:31

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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.

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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.
    – user32479
    Jan 10, 2016 at 22:35
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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, 2016 at 3:51
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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, 2016 at 6:08
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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, 2016 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, 2016 at 14:21
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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.

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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, 2016 at 17:19
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    People actually are surprisingly rational when they spend their money. [citation needed]
    – Zenadix
    Jan 12, 2016 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, 2016 at 16:23
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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, 2016 at 16:52
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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, 2016 at 14:37
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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.

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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, 2016 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, 2016 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.)

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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, 2016 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, 2016 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, 2016 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).
    – user12515
    Jan 11, 2016 at 17:33
  • that sounds like Pascal's wager.
    – stephenbayer
    Jan 13, 2016 at 20:39
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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.)

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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, 2016 at 6:09
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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, 2016 at 6:34
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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, 2016 at 10:37
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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, 2016 at 12:04
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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, 2016 at 14:21
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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.

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  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.

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  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.

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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, 2016 at 1:41
  • @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, 2016 at 2:12
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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.

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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, 2016 at 3:54
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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, 2016 at 6:04
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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, 2016 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, 2016 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

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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, 2016 at 9:54
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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, 2016 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
    Jan 11, 2016 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, 2016 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, 2016 at 15:28
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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.

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  • I'd love to hear a counter-argument.
    – Peter - Reinstate Monica
    Jan 13, 2016 at 9:39
  • 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, 2016 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, 2016 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.

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  • 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, 2016 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, 2016 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, 2016 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.
    – user9822
    Jan 13, 2016 at 2:51
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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.

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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, 2016 at 16:39
  • How many tickets are outstanding?
    – Pete Becker
    Jan 10, 2016 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, 2016 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, 2016 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, 2016 at 5:29
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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.

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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 4, 2016 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 4, 2016 at 4:45
  • @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 4, 2016 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 4, 2016 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 4, 2016 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...

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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 3, 2016 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.

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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, 2016 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'.

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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, 2016 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. :)

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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!

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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)

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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.

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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, 2016 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, 2016 at 17:42

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