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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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20 Answers20

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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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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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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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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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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.)

JoeTaxpayer
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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.

Jasper
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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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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

JoeTaxpayer
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Mawg
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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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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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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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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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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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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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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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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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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.

RedScourge
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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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