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Ask HN: Expected value of a $2 powerball ticket is now $4.10. Is it worth it?

With a jackpot of $1.2 billion, and odds of winning at 1:292201338, the expected pre-tax value of a $2 ticket is $4.10 before taxes, and still profitable after taxes. If so, are you justifying it with the expected value, enjoyment, or something else? It struck me that this is similar to the arguments for spending time/money/energy preventing "S-risks", like the very unlikely but very bad possibility of hostile AI takeover or other things talked about in the LessWrong community. Here we have an example of a precisely known magnitude of good that could come about, and a precisely known probability of it happening.

mrep 4y ago
600 million cash though so really $2 and that is before taxes which will take about half so more like $1
mikkergp 4y ago
Except for ticket splitting right? It’s not exact precision. There’s a non zero chance you’re jackpot is only 1/2 or 1/3rd of the value.
- PaulHoule 4y ago
  For optimal play you want to pick a number which is not popular among other players. On one hand there are numbers (like the lottery winner from last week!) that people overbet. There are other numbers that are systematically underbet. The "minimax" solution comes very close to having the lottery pick a random number for you, which a high fraction of players (maybe 70%) do. Since so many people are playing quick picks already the benefits of picking an systematically underbet number are greatly diminished, so you might just bet a quick pick yourself.
elmerfud 4y ago
I buy them as my "retire earlier" plan. I have disposable income and it doesn't hurt or even make a noticable impact to my lifestyle to spend $40 on it.
elefantastisch 4y ago
We have an even better example of mistaking the numbers for the whole picture. Becoming famous for winning obscene amounts of money greatly increases your chance of being harassed, murdered, or driven to mental breakdown. The expected value of a losing ticket (a chance to fantasize, enjoyment, etc.) is much higher than the expected value of a winning ticket (a lifetime of being hounded for money in increasingly desperate and violent ways).
pestatije 4y ago
I don't want one chance to win $1.2b...i want 1000 chances to win $1.2m...and there is no lotto in the world offering that.
- Axsuul 4y ago
  A startup?
  - AnimalMuppet 4y ago
    Is there an automated way to buy tickets (online, or whatever)? A startup could hire someone to buy tickets all day, but that would get tedious...
    - thoughtpalette 4y ago
      You can "subscribe" on at least the IL Lottery site to "auto buy" tickets.
    - evilDagmar 4y ago
      There's been a number of instances where people did precisely that--they just sent a bunch of taskrabbits out to buy large numbers of tickets in lots of different locations. Very time-consuming, but still profitable.
    - terran57 4y ago
      there's an app called JackPocket
    - pytonslange 4y ago
      I think I just understood that the official outlet for lottery tickets physical stores. Is that really so? As a European I am kind of baffled by this, as we (at least in my country) have had an official app for buying tickets for the state lottery and sports betting for ages. What is the reason for that? Do the players only trust hard physical paper numbers? Btw, the euro jackpot is about €120m, but still far from profitable.
  - 4y ago
tipsytoad 4y ago
I wrote an article on an alternative ev metric[1] which takes into account the player having a discrete bankroll (rather than infinite assumed by ev). https://tompollak.me/blog/ev-v-eg/ https://tompollak.me/blog/ev-v-eg/
mint2 4y ago
Expected value here should include some measure or utility. Like “expected value of useful money.” After the lottery is over 20 million, how much difference does it really make whether it’s 100M, 500M, or 1B. If there were 1k chances at 1M then I might consider a ticket. But 1 chance at 1B is about as good as 1 chance at 1M. Unless you’re pooling with several thousand people, Do your expected value at the point where the money’s impact levels off.
enahs-sf 4y ago
I thought the insurance companies that underwrite the lotto start buying tickets at this point?
toast0 4y ago
The sum of the annunity payments is not a very useful number. You really need to use the cash value or construct your own present value of the sequence of payments. Ignoring taxes is also a mistake, because even annuitized, most of the winnings are going to be taxed at the top rate which is approximately 40%. That said, I'm willing to ignore the tax issue, and I'll generate some numbers and pay my $2 for Wednesday's drawing. The economic utility of pissing away $2 on this may approach 40 cents.
- toast0 4y ago
  As an update, I wasn't able to get my $2 in, but the numbers I generated wouldn't have won. $2 saved to piss away for Saturday.
- extraduder_ire 4y ago
  It's probably just because I'm not used to it, but taxing winnings from gambling never made much sense to me. Especially if it's run as a government monopoly. Taking a cut on the house-side seems way more stable and assured to me, and the numbers are more closely related to what you get. It's a shame the US taxes citizens wherever they are on earth, or I'd expect the done thing to be leaving the country to avoid those taxes, since the money saved would certainly cover it. I'm sure there's a few accountants around who could come up with a tax minimization scheme that's well worth their cost though. Not here though, national lottery/euromillions are tax free and paid out in one lump sum.
  - toast0 4y ago
    > Especially if it's run as a government monopoly. It's a state-run government monopoly. The federal government taxes the winnings, but can't tax the state revenue, as state and local government are tax exempt. The company that runs the lottery is taxable, of course, but they're (hopefully) not getting the lions share of the house edge. I know California doesn't charge their income tax on winnings from their lottery though. The problem with tax minimization on lottery winnings is that you need to do the work before the tickets are valuable. If you make a habit of buying your lottery tickets in a tax advantaged account, then ok; but if you try to move it after it's a winning ticket, that's not going to work well.
skue 4y ago
Your odds of winning are no higher with a larger jackpot, and it is only profitable if you win. And unless you plan to invest in a massive number of tickets, the Law of Large Numbers doesn’t apply.
giantg2 4y ago
People don't play the lottery because they crunched the numbers (and any number justifying it are likely gamed). People play the lottery in the hope that our shitty lives will be fixed by money.
- thenoblesunfish 4y ago
  That's a little harsher than I'd put it - I'd just say that gambling itself has entertainment value to people, which explains why they will, in expectation, pay to do it (see: state of Nevada).
  - giantg2 4y ago
    The thing that makes it entertaining is the hope of winning.
thenoblesunfish 4y ago
Someone who knows more about probability say more. Is expectation the right thing to talk about here? I never heard it said explicitly, but if you're talking about expectation, there's some underlying idea that you're going to do the thing many times and add/average the results, right?
- constantcrying 4y ago
  Exactly. That is the frequentist interpretation of probability. In this case it is easy to calculate. Probability to win x Money gained + Probability to loose x money lost.
jwalton 4y ago
With the odds of winning at 1:292201338, the expected value of a single powerball ticket is still very much $0. You would need to buy hundreds of millions of tickets to see a positive expected value.
- SideQuark 4y ago
  That's not how expected value works. The expected value is positive. You're confusing expected value with volatility.
DerekBickerton 4y ago
The lotto is a tax on the poor. If you are any way mathematically minded, you can calculate the odds and compare it with getting struck by lightning on the top of Mount Everest or being abducted by a UFO. The folly of the lotto still amazes me.
- thunky 4y ago
  > The lotto is a tax on the poor. This is a common saying, but it's a backhanded exaggeration. Maybe more accurate to say it's a tax on those that want to be rich.
  - soulofmischief 4y ago
    I can think of few would like to be rich more than the poor.
    - thunky 4y ago
      Yes but they're not the only ones that play the lotto, not by a long shot. The point is that going out of your way to say that poor people are the only ones dumb enough to buy lotto tickets is unnecessary. Nobody calls sports betting a tax on the middle class.
      - soulofmischief 4y ago
        It's called a tax on the poor because they're the ones who often desire it most, while also being the ones who can least afford the luxury of pissing away money. At some point you have to admit there is a deception taking place.
Larrikin 4y ago
The lottery should only be played if the amount of money you spend on it could be thrown in a fireplace and not have any effect on your life. If you want to do something more fun than watch the money burn, then play the lotto and have fun. There's no need to have condescending attitudes about it and definitely don't try to game out the ridiculous odds to justify spending money you wouldn't spend otherwise.
deleted 4y ago
[deleted]
1123581321 4y ago
It is negative expected value because of pot splitting. It came closest to positive EV when the jackpot was around $200-300MM. As interest increases, the number of tickets sold per draw forces the EV downward. You also need to factor the present value if basing on the annuity and not the lump sum.
constantcrying 4y ago
Even if the actual EV (money you spend vs. money ultimately in your bank account) were positive, it still might be a bad bet. If you look at it as an investment, at the very least you have to take into consideration risk and opportunity costs. The risk is enormous, to get 50/50 odds you would need to invest 400 Million. Imagine what else you could do with that money instead, without it being a 50% chance of loosing everything.
8bitsrule 4y ago
Those over 50M jackpots should be divided between n=jackpot/50M winners. (Keep picking matching numbers out of the pool of sold tickets.) That'd keep things more interesting.

mrep 4y ago

600 million cash though so really $2 and that is before taxes which will take about half so more like $1

mikkergp 4y ago

Except for ticket splitting right? It’s not exact precision. There’s a non zero chance you’re jackpot is only 1/2 or 1/3rd of the value.

PaulHoule 4y ago
For optimal play you want to pick a number which is not popular among other players. On one hand there are numbers (like the lottery winner from last week!) that people overbet. There are other numbers that are systematically underbet. The "minimax" solution comes very close to having the lottery pick a random number for you, which a high fraction of players (maybe 70%) do. Since so many people are playing quick picks already the benefits of picking an systematically underbet number are greatly diminished, so you might just bet a quick pick yourself.

elmerfud 4y ago

I buy them as my "retire earlier" plan. I have disposable income and it doesn't hurt or even make a noticable impact to my lifestyle to spend $40 on it.

elefantastisch 4y ago

We have an even better example of mistaking the numbers for the whole picture. Becoming famous for winning obscene amounts of money greatly increases your chance of being harassed, murdered, or driven to mental breakdown. The expected value of a losing ticket (a chance to fantasize, enjoyment, etc.) is much higher than the expected value of a winning ticket (a lifetime of being hounded for money in increasingly desperate and violent ways).

pestatije 4y ago

I don't want one chance to win $1.2b...i want 1000 chances to win $1.2m...and there is no lotto in the world offering that.

Axsuul 4y ago
A startup?
- AnimalMuppet 4y ago
  Is there an automated way to buy tickets (online, or whatever)? A startup could hire someone to buy tickets all day, but that would get tedious...
  - thoughtpalette 4y ago
    You can "subscribe" on at least the IL Lottery site to "auto buy" tickets.
  - evilDagmar 4y ago
    There's been a number of instances where people did precisely that--they just sent a bunch of taskrabbits out to buy large numbers of tickets in lots of different locations. Very time-consuming, but still profitable.
  - terran57 4y ago
    there's an app called JackPocket
  - pytonslange 4y ago
    I think I just understood that the official outlet for lottery tickets physical stores. Is that really so? As a European I am kind of baffled by this, as we (at least in my country) have had an official app for buying tickets for the state lottery and sports betting for ages. What is the reason for that? Do the players only trust hard physical paper numbers? Btw, the euro jackpot is about €120m, but still far from profitable.
- 4y ago

tipsytoad 4y ago

I wrote an article on an alternative ev metric[1] which takes into account the player having a discrete bankroll (rather than infinite assumed by ev). https://tompollak.me/blog/ev-v-eg/ https://tompollak.me/blog/ev-v-eg/

mint2 4y ago

Expected value here should include some measure or utility. Like “expected value of useful money.” After the lottery is over 20 million, how much difference does it really make whether it’s 100M, 500M, or 1B. If there were 1k chances at 1M then I might consider a ticket. But 1 chance at 1B is about as good as 1 chance at 1M. Unless you’re pooling with several thousand people, Do your expected value at the point where the money’s impact levels off.

enahs-sf 4y ago

I thought the insurance companies that underwrite the lotto start buying tickets at this point?

toast0 4y ago

The sum of the annunity payments is not a very useful number. You really need to use the cash value or construct your own present value of the sequence of payments. Ignoring taxes is also a mistake, because even annuitized, most of the winnings are going to be taxed at the top rate which is approximately 40%. That said, I'm willing to ignore the tax issue, and I'll generate some numbers and pay my $2 for Wednesday's drawing. The economic utility of pissing away $2 on this may approach 40 cents.

toast0 4y ago
As an update, I wasn't able to get my $2 in, but the numbers I generated wouldn't have won. $2 saved to piss away for Saturday.
extraduder_ire 4y ago
It's probably just because I'm not used to it, but taxing winnings from gambling never made much sense to me. Especially if it's run as a government monopoly. Taking a cut on the house-side seems way more stable and assured to me, and the numbers are more closely related to what you get. It's a shame the US taxes citizens wherever they are on earth, or I'd expect the done thing to be leaving the country to avoid those taxes, since the money saved would certainly cover it. I'm sure there's a few accountants around who could come up with a tax minimization scheme that's well worth their cost though. Not here though, national lottery/euromillions are tax free and paid out in one lump sum.
- toast0 4y ago
  > Especially if it's run as a government monopoly. It's a state-run government monopoly. The federal government taxes the winnings, but can't tax the state revenue, as state and local government are tax exempt. The company that runs the lottery is taxable, of course, but they're (hopefully) not getting the lions share of the house edge. I know California doesn't charge their income tax on winnings from their lottery though. The problem with tax minimization on lottery winnings is that you need to do the work before the tickets are valuable. If you make a habit of buying your lottery tickets in a tax advantaged account, then ok; but if you try to move it after it's a winning ticket, that's not going to work well.

skue 4y ago

Your odds of winning are no higher with a larger jackpot, and it is only profitable if you win. And unless you plan to invest in a massive number of tickets, the Law of Large Numbers doesn’t apply.

giantg2 4y ago

People don't play the lottery because they crunched the numbers (and any number justifying it are likely gamed). People play the lottery in the hope that our shitty lives will be fixed by money.

thenoblesunfish 4y ago
That's a little harsher than I'd put it - I'd just say that gambling itself has entertainment value to people, which explains why they will, in expectation, pay to do it (see: state of Nevada).
- giantg2 4y ago
  The thing that makes it entertaining is the hope of winning.

thenoblesunfish 4y ago

Someone who knows more about probability say more. Is expectation the right thing to talk about here? I never heard it said explicitly, but if you're talking about expectation, there's some underlying idea that you're going to do the thing many times and add/average the results, right?

constantcrying 4y ago
Exactly. That is the frequentist interpretation of probability. In this case it is easy to calculate. Probability to win x Money gained + Probability to loose x money lost.

jwalton 4y ago

With the odds of winning at 1:292201338, the expected value of a single powerball ticket is still very much $0. You would need to buy hundreds of millions of tickets to see a positive expected value.

SideQuark 4y ago
That's not how expected value works. The expected value is positive. You're confusing expected value with volatility.

DerekBickerton 4y ago

The lotto is a tax on the poor. If you are any way mathematically minded, you can calculate the odds and compare it with getting struck by lightning on the top of Mount Everest or being abducted by a UFO. The folly of the lotto still amazes me.

thunky 4y ago
> The lotto is a tax on the poor. This is a common saying, but it's a backhanded exaggeration. Maybe more accurate to say it's a tax on those that want to be rich.
- soulofmischief 4y ago
  I can think of few would like to be rich more than the poor.
  - thunky 4y ago
    Yes but they're not the only ones that play the lotto, not by a long shot. The point is that going out of your way to say that poor people are the only ones dumb enough to buy lotto tickets is unnecessary. Nobody calls sports betting a tax on the middle class.
    - soulofmischief 4y ago
      It's called a tax on the poor because they're the ones who often desire it most, while also being the ones who can least afford the luxury of pissing away money. At some point you have to admit there is a deception taking place.

Larrikin 4y ago

The lottery should only be played if the amount of money you spend on it could be thrown in a fireplace and not have any effect on your life. If you want to do something more fun than watch the money burn, then play the lotto and have fun. There's no need to have condescending attitudes about it and definitely don't try to game out the ridiculous odds to justify spending money you wouldn't spend otherwise.

deleted 4y ago

[deleted]

1123581321 4y ago

It is negative expected value because of pot splitting. It came closest to positive EV when the jackpot was around $200-300MM. As interest increases, the number of tickets sold per draw forces the EV downward. You also need to factor the present value if basing on the annuity and not the lump sum.

constantcrying 4y ago

Even if the actual EV (money you spend vs. money ultimately in your bank account) were positive, it still might be a bad bet. If you look at it as an investment, at the very least you have to take into consideration risk and opportunity costs. The risk is enormous, to get 50/50 odds you would need to invest 400 Million. Imagine what else you could do with that money instead, without it being a 50% chance of loosing everything.

8bitsrule 4y ago

Those over 50M jackpots should be divided between n=jackpot/50M winners. (Keep picking matching numbers out of the pool of sold tickets.) That'd keep things more interesting.