# Does it make sense to play the Powerball now?

**URL:** <https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692>\
**Category:** Factual Questions\
**Created:** [January 10, 2016, 4:26pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692 "2016-01-10T16:26:33Z")\
**Posts on this page:** 16\
**Page:** 4

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**Author:** ![md2000](https://avatars.discourse-cdn.com/v4/letter/m/73ab20/32.png) [@md2000](https://boards.straightdope.com/u/md2000)\
**Post date:** [January 12, 2016, 1:42pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/61 "2016-01-12T13:42:14Z")

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As I’ve said, your chances of winning go up tremendously when you buy that first ticket. After that, the law of diminishing returns sets in. Your 100th ticket doesn’t better your odds by much.

IIRC more people win $1M than are struck by lightning each year - so unless you are Ranger Smith, one ticket every so often instead of a Starbucks can’t hurt.

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**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\
**Post date:** [January 12, 2016, 2:32pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/62 "2016-01-12T14:32:22Z")

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> [@sbunny8](#):
>
> And I hope you agree that, if twice as many tickets get sold, that would double the expected number of winners, which would cut in half the expected value of each winning ticket.

I’ll disagree. 😃 The more tickets sold, the more money is in the pool.

To work out the details, we need to know how the jackpot rollover works, and what portion of ticket sales goes to the jackpot. Anyone have those numbers?

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**Author:** ![Surreal](https://avatars.discourse-cdn.com/v4/letter/s/eada6e/32.png) [@Surreal](https://boards.straightdope.com/u/Surreal)\
**Post date:** [January 12, 2016, 3:10pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/63 "2016-01-12T15:10:34Z")

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> [@CC](#):
>
> Look, when there are millions and millions of people buying tickets and millions and millions of possible winning numbers. you’re kidding yourself if you think you have anything more than a hope in hell of winning.

I’m pretty sure everyone already knows this.

> [@CC](#):
>
> This stuff about which numbers are more likely to come up, how many people are buying, what numbers get played most often, etc. are just ways of entertaining yourself while you watch your overall wealth go down by a couple of dollars.

Oh noes, some people’s net worth dropping by _two whole dollars_! How will they be able to afford to eat?

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**Author:** ![jtur88](https://avatars.discourse-cdn.com/v4/letter/j/e9c0ed/32.png) [@jtur88](https://boards.straightdope.com/u/jtur88)\
**Post date:** [January 12, 2016, 6:29pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/64 "2016-01-12T18:29:49Z")

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In short, Yes, because my chance of winning remains constant, while the payout keeps increasing. The risk of sharing with a second winner does not offset the many-fold increase in he payout. In fact, it is better to win now for half or even a third of the prize, than to have won the whole thing two weeks ago when it was under $400m. Of the billion or so tickets already bought, a great majority of them are torn up and no longer competing with me, but all that money is still in the jackpot waiting for me.  
Say there is a guy with a deck of cards, and people have been betting, but nobody winning, and the kitty is now up to $200. If you guess the correct card drawn, you get all the money already wagered by losers who have left the game, but your odds are still 52:1. But 75 people are now betting, so you might have to split the prize and only get $100. Still a better bet than it was when he first took out the cards.

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**Author:** ![Hampshire](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hampshire/32/4887_2.png) [@Hampshire](https://boards.straightdope.com/u/Hampshire)\
**Post date:** [January 12, 2016, 9:19pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/65 "2016-01-12T21:19:11Z")

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A 4000 square foot lawn (40x100) contains about 200,000,000 blades of grass.  
So your odds of winning the jackpot are about the same as wandering out onto that lawn and picking the one winning blade of grass.  
Astronomical odds? Sure.  
Would I be willing to drop $20 to wander out there and pick 10 blades of grass?  
Why not. I can spare $20.  
They say the last round 75% of the possible combinations were purchased. So there was a 75% chance that SOMEBODY was going to win.  
Crappy odd that YOU will win sure, but that’s some good odds that SOMEBODY will win.  
I’d feel sort of left out if I wasn’t in on the opportunity at least.

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**Author:** ![Hail\_Ants](https://avatars.discourse-cdn.com/v4/letter/h/dc4da7/32.png) [@Hail\_Ants](https://boards.straightdope.com/u/Hail_Ants)\
**Post date:** [January 12, 2016, 9:32pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/66 "2016-01-12T21:32:42Z")

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> [@pulykamell](#):
>
> Given that the five match prize is a flat $1 mil, no matter how many winners, yeah, I’d probably be a bit upset if matching all six paid put at much less.

Whatever, but regardless of anything if someone handed me a check for (after taxes) around $100,000 I really don’t think ‘upset’ would be the word I’d use (unless I was a compulsive gambler)…

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**Author:** ![Spiderman](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/spiderman/32/230_2.png) [@Spiderman](https://boards.straightdope.com/u/Spiderman)\
**Post date:** [January 13, 2016, 3:46am UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/67 "2016-01-13T03:46:56Z")

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> [@The\_Universe\_Lashes\_Out](#):
>
> According to this [Business Insider](http://www.businessinsider.com/powerball-lottery-expected-value-jan-9-draw-2016-1) article, the jackpot needs to be almost $1.6 billion before the expectation turns positive. And that’s without factoring in the chances of splitting the prize. I know it’s up over a billion, but still not quite at the level that would even come close to making sense, economically speaking.

At this point, with almost a day to go it’s officially at 1.5 billion so it might just get there. Though I’ll disagree with you on one point, I don’t think the lottery ever makes sense, economically speaking. However, spending $2 on the lottery means I buy one less soda, which is bad for me so there’s a health benefit to playing some nominal amount…& if I win, I’ll be able to employee people to drink soda for me.

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**Author:** ![opossum](https://avatars.discourse-cdn.com/v4/letter/o/2bfe46/32.png) [@opossum](https://boards.straightdope.com/u/opossum)\
**Post date:** [January 13, 2016, 6:57am UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/68 "2016-01-13T06:57:03Z")

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> [@Spiderman](#):
>
> At this point, with almost a day to go it’s officially at 1.5 billion so it might just get there. Though I’ll disagree with you on one point, I don’t think the lottery ever makes sense, economically speaking. However, spending $2 on the lottery means I buy one less soda, which is bad for me so there’s a health benefit to playing some nominal amount…& if I win, I’ll be able to employee people to drink soda for me.

I am both excellent at and passionate about drinking soda, and if you win I’d like do be considered. Could I work from home?

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**Author:** ![Tired\_and\_Cranky](https://avatars.discourse-cdn.com/v4/letter/t/a87d85/32.png) [@Tired\_and\_Cranky](https://boards.straightdope.com/u/Tired_and_Cranky)\
**Post date:** [January 13, 2016, 3:38pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/69 "2016-01-13T15:38:57Z")

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> [@sbunny8](#):
>
> And I hope you agree that, if twice as many tickets get sold, that would double the expected number of winners, which would cut in half the expected value of each winning ticket.
> 
> If the jackpot gets bigger but the number of tickets sold increases proportionately, that doesn’t improve your expected winnings.

Thanks **sbunny8**. You pointed out something that should have been obvious to me but wasn’t; increasing the size of the pot by selling more tickets doesn’t increase the expected value because it makes sharing the pot more likely. So the expected value of a ticket this week is still less than $2. I guess the prize has to roll over again for it to have any chance to break $2.

And, for what it’s worth, I read that 95% of the tickets sold last week were drawn randomly by the computer. That’s likely close to a perfect random distribution so your simplifying assumption seems pretty valid.

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**Author:** ![gazpacho](https://avatars.discourse-cdn.com/v4/letter/g/6f9a4e/32.png) [@gazpacho](https://boards.straightdope.com/u/gazpacho)\
**Post date:** [January 13, 2016, 3:51pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/70 "2016-01-13T15:51:03Z")

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These guys calculate that with 1.4 billion payout and some assumptions about the number of tickets sold that the expected payout for a $2 ticket is $1.32.

> **[Is It Mathematically Rational To Purchase A Powerball Ticket?](https://thefederalist.com/2016/01/12/with-the-jackpot-at-1-4-billion-is-it-finally-rational-to-purchase-a-powerball-ticket/)**
>
> With the Powerball jackpot now at $1.4 billion, is it finally mathematically rational to purchase a Powerball lottery ticket?

Five thirty eight has been writing some articles about the lottery as well.

> **[Everyone Is Freaking Out About The $1.5 Billion Powerball, And The Stats Agree](https://fivethirtyeight.com/features/billion-dollar-powerball-lottery/)**
>
> Well, this is it: the first North American lottery over $1 billion. With an advertised jackpot of $1.5 billion, and only five other Powerball lotteries north of…

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**Author:** ![sbunny8](https://avatars.discourse-cdn.com/v4/letter/s/c77e96/32.png) [@sbunny8](https://boards.straightdope.com/u/sbunny8)\
**Post date:** [January 13, 2016, 4:43pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/71 "2016-01-13T16:43:29Z")

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> [@Tired\_and\_Cranky](#):
>
> Thanks **sbunny8**. You pointed out something that should have been obvious to me but wasn’t; increasing the size of the pot by selling more tickets doesn’t increase the expected value because it makes sharing the pot more likely. So the expected value of a ticket this week is still less than $2. I guess the prize has to roll over again for it to have any chance to break $2.

The ideal condition would be a rollover followed by a sudden lack of interest on the part of ticket buyers. That’s a pretty rare circumstance.

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**Author:** ![Surreal](https://avatars.discourse-cdn.com/v4/letter/s/eada6e/32.png) [@Surreal](https://boards.straightdope.com/u/Surreal)\
**Post date:** [January 13, 2016, 5:45pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/72 "2016-01-13T17:45:02Z")

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> [@gazpacho](#):
>
> These guys calculate that with 1.4 billion payout and some assumptions about the number of tickets sold that the expected payout for a $2 ticket is $1.32.

I don’t see how this information is useful as it doesn’t represent any real-world outcome. You’d have to buy millions of tickets to have any confidence in the $1.32 average payout.

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**Author:** ![gazpacho](https://avatars.discourse-cdn.com/v4/letter/g/6f9a4e/32.png) [@gazpacho](https://boards.straightdope.com/u/gazpacho)\
**Post date:** [January 13, 2016, 6:42pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/73 "2016-01-13T18:42:36Z")

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> [@Surreal](#):
>
> I don’t see how this information is useful as it doesn’t represent any real-world outcome. You’d have to buy millions of tickets to have any confidence in the $1.32 average payout.

You are probably correct that the expected value is not very useful. But the original question is that now that the jackpot is so high does it make sense to buy lottery tickets. This to me implies determining if the odds are in your favor. And they still are very much not in your favor.

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**Author:** ![sbunny8](https://avatars.discourse-cdn.com/v4/letter/s/c77e96/32.png) [@sbunny8](https://boards.straightdope.com/u/sbunny8)\
**Post date:** [January 13, 2016, 7:28pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/74 "2016-01-13T19:28:43Z")

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> [@Surreal](#):
>
> I don’t see how this information is useful as it doesn’t represent any real-world outcome. You’d have to buy millions of tickets to have any confidence in the $1.32 average payout.

Expected value gives you a measure of whether a game is tilted in favor of the player or the house (and by how much) or if the game is fair. It also gives you a way to compare one game to another when deciding which game to play.

In this case, determining that the expected value of the ticket is $1.32 and it costs $2.00 to buy the ticket shows that this game heavily favors the house. Contrast that with a roulette wheel which has 38 spaces numbered 1-36, 0, and 00. A $2 bet on any single number pays out $72 and the probability of winning is 1/38 so the expected value is approximately $1.89. This shows that roulette is a better choice than the lottery, because 1.89\>1.32, although both of them are less than 2.00 (the cost of the ticket), so both games favor the house.

And this information may not seem useful to you but it is certainly useful to the house because they WILL sell millions of tickets.

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**Author:** ![Gatopescado](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/gatopescado/32/5711_2.png) [@Gatopescado](https://boards.straightdope.com/u/Gatopescado)\
**Post date:** [January 13, 2016, 8:10pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/75 "2016-01-13T20:10:48Z")

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No.

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<div class="post-metadata">

**Author:** ![sbunny8](https://avatars.discourse-cdn.com/v4/letter/s/c77e96/32.png) [@sbunny8](https://boards.straightdope.com/u/sbunny8)\
**Post date:** [January 14, 2016, 3:46pm UTC](https://boards.straightdope.com/t/does-it-make-sense-to-play-the-powerball-now/742692/76 "2016-01-14T15:46:35Z")

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> [@sbunny8](#):
>
> Short answer, No. Don’t buy the ticket.
> 
> Don’t let the size of the jackpot fool you. The bigger the jackpot, the more tickets they sell this week, and the greater the chances that whoever wins will end up splitting it with several other people. In order to calculate the expected value of a ticket, you need to know how many tickets there are.
> 
> Let t= number of tickets. p= size of jackpot. n= number of unique combinations. The expected number of winning tickets then would be t/n, so the actual payout is expected to be p(n/t). And the chance of your ticket winning is 1/n. If c is the cost of the ticket, the expected value of your ticket E = (1/n)(p)(n/t)-c\*(n-1)/n. For large values of n, this is essentially E = p/t-c. If p is $1.4 billion and c is $2, then E\>0 if and only if t\<700 million.

D’oh! I can’t believe I forgot to account for the fact that when they say that the jackpot is 1.5 billion IT’S NOT REALLY 1.5 BILLION. Let me explain.

Let’s play a game. We’re going to bet on a coin flip. You have to pay me $15 before we flip the coin. If it comes up heads, I keep your $15 and you get nothing. If it comes up tails, I keep your $15 and I pay you $30. Sounds fair so far, right? It’s 50-50 odds and I’m offering you double-or-nothing. But here’s the catch. If you win, I won’t give you the $15 all at once; I’m only going to give you $1 a year for 30 years. However, I’ll also give you the option to take a lump sump of $18.60 right now. What do you think of the game now? Does it still sound fair?

Put it another way… Right now you can go to a bank and buy yourself an annuity. If you want an annuity that pays $1,000 per year for the next 30 years, that will only cost you $18,600 right now. So tell me, how much is that annuity worth? It’s worth $18,600. Anyone who claims that it’s worth $30,000 is fudging the truth.

When the lottery says the jackpot is $1.5 billion, what they REALLY mean is that all the winners (three of them in this case) will each get annuities that pay $16,666,667 per year for the next 30 years. How much is that annuity worth right now? $310 million. In accounting terms, that’s called the Present Value.

So now let’s readjust the numbers I posted earlier. Expected value is E = p/t-c. If they say the jackpot is $1.5 billion then that means p is really 62% of $1.5 billion, which is $930 million. And if c is $2, then E\>0 if and only if t\<465 million.

I speculated that there would be more than 700 million tickets sold. It turns out I estimated too high. Powerball chair Gary Grief told ABC News that 371 million tickets were sold as of Wednesday afternoon. [How Many Powerball Tickets Were Sold For The $1.5 Billion Jackpot? Don't Feel Too Bad If You Lost](http://www.bustle.com/articles/135525-how-many-powerball-tickets-were-sold-for-the-15-billion-jackpot-dont-feel-too-bad-if)

So E = $930m/371m-$2 which comes out to about 51 cents, which means that (on average) Powerball ticket buyers this week got back $2.51 for every $2.00 spent. That’s a positive return on investment. But remember that we didn’t know in advance how many tickets would be sold. Hindsight is 20/20.

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