A ticket costs $1000. Each ticket has a 1 in 1,000 chance of winning $1 billion. How many do you buy?

``` wrote:

Also, my intention was that each ticket has an independent chance at winning--for each ticket, a 1000-side die is rolled, and if it comes up 69, you get a $billion...

Can someone explain to me, with this description, how you wouldn’t be guaranteed to win if you bought all 1,000 numbers? That was my impression under the OPs first post and has only been fortified now.. but I’ll admit I’m not the sharpest tool in the shed.

Along a similar vein, how can you create a lottery with a 1-1000 chance of winning, but have more than 1000 combinations (or numbers).

notafan wrote:

``` wrote:

Also, my intention was that each ticket has an independent chance at winning--for each ticket, a 1000-side die is rolled, and if it comes up 69, you get a $billion...

Can someone explain to me, with this description, how you wouldn’t be guaranteed to win if you bought all 1,000 numbers? That was my impression under the OPs first post and has only been fortified now.. but I’ll admit I’m not the sharpest tool in the shed.

Along a similar vein, how can you create a lottery with a 1-1000 chance of winning, but have more than 1000 combinations (or numbers).

If a surgical procedure has a one in a thousand chance of a particular complication, this doesn't mean that a group of 1,000 patients is guaranteed to have one and only one occurrence of the given complication. Each person has a 1 in 1,000 chance, but each person's odds are independent of other patient's outcomes.

Same idea could be applied to a lottery if done with independent odds. You don't even need to base the lottery on giving out lottery numbers and then drawing a winning number. You could just have a black box that randomly assigns "win" or "lose" to each participant and, in the long run, assigns "win" to one out of every thousand participants on average.

This is an interesting exercise. My answer would depend on how easy it would be to get loans - I haven't gone through that process before so am not familiar with it. Going through a few scenarios:

- No loan available: I would buy around 600 tickets. This gives me a 45.2% chance of winning at least once (winning multiple times is basically the same as winning once in my mind, so not worried about that). This is around the sweet spot for me since I would still have enough savings to be fine if I didn't win. If I bought 700 tickets that only gets my chances to 50.4% and has a much bigger impact on life in the next couple years based on how much savings I have left.

- If I can get a moderately sized loan at a reasonable rate, I probably buy roughly 1000 tickets with some split between cash and the loan. This gets me to 63.3%.

It's interesting to chart out the percent chance of winning by the number of tickets. You get a good return on 200 tickets (18.1%) and 300 tickets (25.9%) but it starts getting tougher after that. 400 just gets you to 33%, by 600 you're at 45.2%, and by the time you get to 1000 it's only another 3.8% incremental chance of winning for that 100. Still a very positive expected value, but not nearly the same ROI as before.

Also this is easy to replicate in Excel - just use the formula below and adjust "A1" to be the # of tickets

=1-(0.999)^(A1)

I would buy as many as I could until I eventually win. Once I win I would buy more with that money.

If there are only 5,000 tickets, and five of them are winners, it makes sense to borrow $5M and buy all of them, right?

So what difference does it make if there are 10,000, 100,000 or a million tickets? Unless the total number is infinity, you can always win more than you paid. Who is going to give you the loan for that? If you agree to split the winning, you won't find any problem with finding a lender.

a closer reading of the rules disallows much of what you guys are saying.

You have to buy the tickets all at once, and there is no going along buying until you win. So, it is a one shot deal. One and done.

Also cannot go in with anyone else since OP said no discussing with anyone until you win.

Also, it is possible to buy 1000 tickets and still not win. There are an infinite number of tickets available, and no guarantee how many winning tickets are in existence.

Hope this helps.

OP are you asking if we have access to 1m liquid?

It's a stretch, but sure

400k out of an existing HELOC, pull 500k out of ROTHs [penalty free] and dip 100k out of taxable savings/brokerage or sell some RSUs whichever has preferential taxes.

That's the 1st million without touching 401ks or selling anything. But yea I'm not rich enough to have 1m in a taxable account/brokerage.

Obviously this is just a lesson in 'do u have a million fam', so congrats if you do as this is free money. That would suck to put up 500k or something for only a 50:50 shot at a bill and to come up empty. Still obviously worth it but I'm covering the field as that's an option.

It becomes a little more interesting [but not very] if there's a hypothetical where you cant cover the field.

re-reading, we are not limited to 1000 tickets, ok terrific.

I liquidate my entire net worth in this hypothetical and borrow as much as they'll let me. I can probably come up with a little over 5m, for an expected payout of 4-6b

yes yes yes yes wrote:

a closer reading of the rules disallows much of what you guys are saying.

You have to buy the tickets all at once, and there is no going along buying until you win. So, it is a one shot deal. One and done.

Also cannot go in with anyone else since OP said no discussing with anyone until you win.

Also, it is possible to buy 1000 tickets and still not win. There are an infinite number of tickets available, and no guarantee how many winning tickets are in existence.

Hope this helps.

Right. I overlooked the part that you cannot tell it to anyone in advance. So you cannot borrow money with the promise of splitting the winning.

While interesting, there's no possible objective answer to this. It all depends on whether the poster can throw away the better part of a million bucks and remain standing.

This applies in real life to Powerball. If you only bought tickets for the extremely high jackpots, you'd have a positive expectation, but it would probably take you more than one lifetime to get there.

So for this, I will buy into the mutual fund that buys and cashes these tickets, and quickly because whoever has a few million around is going to recycle their winnings into more tickets until they're either sold out or the lottery is bankrupt.

Kudos, this is actually a pretty interesting question. I thought it was going to be about math, but the math involved is trivial. It's actually about risk.

I could probably buy 50 tickets with negligible pain, 250 with moderate pain (i.e. substantial early withdrawal penalties), and mayyyybe 500 with substantial pain. So even in the most extreme scenario, I'm betting on something worse than a coin toss. So I'd buy zero.

Two possible exceptions. I do have a rich friend who trusts me. In this imaginary game, I could imagine asking him to back me for, say, $2 million without telling him the details. In reality, I wouldn't do that, because he's not rich enough to handle an outright loss of $2 million without serious pain. And with longer than a week to raise money, I could conceivably extract close to a million in equity from my house. But, for better or worse, I doubt I'd actually do that. The consequences of failure at that point, with a family and young kids, are just too extreme, relative to a life that's currently as comfortable as I need.

Anyway, certainly thought-provoking.

I don't buy any because I have a billion in intrinsic value.

3.14 all over your face.

If you buy 10 tickets your chance of winning is about 1%. If you buy 100 tickets it's about 10%. If you buy 1000 tickets it's about 64%. If you buy 2000 it's about 87%.

692 tickets is the 50/50 chance point, if anyone is interested.

None. Too expensive.

If you can win more than once then I’d probably go 20,000. If not, 7,500.

hedge fund manager wrote:

If you can win more than once then I’d probably go 20,000. If not, 7,500.

To be clear, then you are committing to buying as much as 20 million dollars of tickets. And to be clear, one would assume that is your money. not that of your hedge fund shareholders.

So, I would gather, you are either very very financially endowed personally, or prone to be a bit cavalier with other people's money.

notafan wrote:

Can someone explain to me, with this description, how you wouldn’t be guaranteed to win if you bought all 1,000 numbers? That was my impression under the OPs first post and has only been fortified now.. but I’ll admit I’m not the sharpest tool in the shed.

I see the confusion here. Let me explain, there is only one number for which if you roll it, you get a billion dollars, for example 69. For every 1,000 you pay, you get another ROLL, not another number. So you would be correct that if you bought 1,000 numbers, you’d win it 100%, but in this scenario you just get another roll.

Also, I’ve seen some questionable math on here so let me set the record, if you buy 1,000 tickets, you have a 63% chance of getting at least one ticket that hits the jackpot. It’s hard to logically understand how it isn’t 50% so I’ll explain it in English, and then show the math. Basically it would be 50% if “winning” only included hitting the jackpot once. By the 1000th roll, the probability is 50% that you have hit it exactly once already. But we also can include hitting it multiple times in the definition of winning . So the probabilities of hitting in twice, and three times, and four times etc all add up to another 13 percent chance, giving us the probability of winning 63%.

The math is calculated like this, what’s the probability of missing the win on every roll? Once you find that, then the opposite of that probability is the probability of anything besides losing every time, ie winning.

So the probability of losing a 1/1000 chance is 999/1000. The chances of losing 1,000 times in a row is (999/1000)^1,000 which is roughly 0.36. 36% chance of losing 1,000 rolls means a 63% chance of doing anything but losing, ie winning, but also winning multiple times.

Also, someone asked about 50% and 75% and 90% but I’m too lazy and it’s considerably easier to calculate going the other way around, so hopefully these are useful -

1million = 1000 tickets = 63% of winning

2 mil = 2000 tickets = 86.5%

5 mil = 5000 tickets = 99.4%

With the kind of money we're talking about to have good odds of winning, I wouldn't feel comfortable doing it unless my chances of winning were 95% (ie. statistically significant).