Lottery expected value

I assume this topic has been discussed before, but I cannot find the analysis, and I pose it now in honor of the $258.5 million (annuity) jackpot won a few days ago in the Powerball lottery:

Given that the jackpot fund of one of the major lotteries (MegaMillions or Powerball) has rolled over a few times, what data would be required and how would it properly be used to calculate whether a ticket purchase actually had a positive expected value (pre-tax)?

I think the following data might be sufficient:
(1) The portion of ticket sales that are used to fund prizes.
(2) The portion of the total prize fund devoted to the jackpot fund.
(3) The estimated lump sum jackpot payout (not the annuity) for the previous drawing, in which the jackpot was not awarded.
(4) The estimated lump sum jackpot payout for the upcoming drawing.
(5) The basic rules of the lottery drawing.

I think all of these items are available from the lottery corporations, though you might have to collect the data during periods leading up to two drawings to get both (3) and (4).

Item (3) gives a figure for the rollover amount. The difference between (3) and (4) gives the expected increase in the jackpot fund, which could be used with (1) and (2) to derive the estimated number of tickets to be purchased for the upcoming drawing. That expected number of entries plus (5) should enable calculation of the probability that the jackpot will be awarded on that drawing, and (I think) it can be assumed that the non-jackpot prize fund will be fully distributed also.

Isn’t that enough info to calculate the expected value of a ticket? What is the formula for the whole kit and caboodle?

Potential complications: There is some holdout to establish the start-up fund for the next jackpot, but that should be balanced by the “old money” that was used to initially start the current jackpot fund. I also remember some discussion that one of the lotteries was going to limit the rate at which the jackpot could rise, to control lottery fever, which would mean that less than the prescribed amount would be added to the jackpot fund, with the excess being held over for the future. That would complicate things, so I think I would ignore that initially.

Any suggestions?

Maybe you missed this article on the Wiz' main site, currently at the top of the "What's new" section:
https://wizardofodds.com/megabucks

There was also this thread:
https://wizardofvegas.com/forum/gambling/slots/1228-megabucks-odds/
In it, the Wiz talks about the above article, prior to posting it to the public.

Yes, I have seen the discussion of the slot machine game. I think there are different factors involved in the lotteries, and certainly different win probabilities. There is also perhaps some difference between (1) a slot machine jackpot growing with each pull and each pull having a chance to win immediately and (2) the lottery with a batch of entries compiled into one drawing in which a winner may or may not be found.

Oh. Lottery, not slots.

Never mind.

Although there may be some quibbling about the math and the methodology, a good rule of thumb is the number 7.

You are 7 times more likely to get killed driving one mile to buy a ticket than you are likely to buy a winning ticket. Also that is based on there only being ONE person who bought the winning combination.

People who see a humungous jackpot and think they can buy the prize through the purchase of every single possible combination will get real tired filling out ticket forms and will be real disappointed to find others sharing the prize that they bought.

So that 1 in 200,000,000. is a bit on the high side for buying a winning jackpot ticket ... and ofcourse your winnings are taxable so ... its better to save your money for a trip to Vegas where you get a much better deal.

Your analysis, of course, is focused on the very poor likelihood of winning the jackpot. My question was in terms of having a positive expected value for a lottery ticket, Cf., not every video poker advantage player is playing only for the possibility of a royal flush on every pull. So at what point is a lottery ticket a favorable wager (in terms of expected value), even if it isn't very likely to make you richer than a rock star? What data is needed for the calculation, and how should the calculation be made?

There have been attempts by investment groups to buy enough tickets to guarantee that they win the lottery. State officials have made it much harder to purchase large numbers of tickets in numerical sequence to prevent actions like this in the future.

It's a poor investment strategy, since you don't know that you will positively purchase every number, and might not get the winning number, and you can't control how many other people might win. The state officials didn't come up with the new rules to save money, but they feel that the game will be cheapened if you average citizen knows these syndicates are operating.

If I was trying to double $100K I would take it to Main Street, and Eastside Cannery and just play the hell out of the free odds on craps. I know that is not the same as trying to win $100's of millions, but the investment consortium to cover the tickets would be very difficult to put together.

Quote: Doc

I think the following data might be sufficient:
(1) The portion of ticket sales that are used to fund prizes.
(2) (3) (4) (5)

I think once you find out the answer to (1) you realize that the house edge is so high it is never going to be a positive expectation, particularly since you can't be sure that only you are the purchaser of the winning combination.

The lotteries raise tax money, usually with some sort of publicity about funding education (never about funding teacher's salaries, just education or schools). This mandated use of the funds is usually fifty percent. Then about half the rest is administrative costs for the lottery, including rebates to vendors. Some portion obviously does go to fund annuities to cover the prizes. An estimate of ticket sales can be derived from the number of free tickets that are awarded.

Once you look at the pittance actually used to fund significant prizes, there is not much use in looking any further to see about positive expectation. Throw in the more demanding rules to qualify for some of the prizes and the fact that the prizes are taxable income and its simply an exercise in futility rather than an exercise in combinatorics.

As to funding prizes: One NY family turned in a winning ticket to the NYS lottery in the name of their two year old son so they would get the lifetime payments for a long time but were shocked to learn that only the two year old would get the money and that would happen only if he lived to be 18 and that all interest on the money would go to the lottery, not them. It takes very little to purchase an annuity to fund the lottery payments.

Lotteries draw customers to convenience stores particularly in poor neighborhoods and boost the sale of alcohol and tobacco. Casanova got rich with the lottery that he established in Paris... he bought eleven ticket-selling franchises!

For what it's worth here is the info I have for Powerball. 50% of the money from sold tickets goes to the prize pool. Of the money in the prize pool, about 65% goes to the jackpot. The jackpot (before or after taxes, lumpsum or annuity depending on your preferance) would have to be more than $161136764 to have a positie EV. (Slightly more if you take out taxes on the lower prizes. Can provide and excel file if you want it.) You also need to take into account the odds of splitting the jackpot, but that is beyond my math skills.

white	red	combinations	probability	pays	return	percent of prize pool
5	1	1	0.000000005121664	161136764	0.82528832124329	65.0577
5	0	38	0.000000194623222	200000	0.038924644418508	7.7849
4	1	270	0.00000138284921	10000	0.013828492096049	2.7657
4	0	10260	0.000052548269965	100	0.005254826996499	1.051
3	1	14310	0.000073291008109	100	0.007329100810906	1.4658
3	0	543780	0.002785058308144	7	0.01949540815701	3.8991
2	1	248040	0.00127037747389	7	0.008892642317232	1.7785
2	0	9425520	0.048274344007833	0	0
1	1	1581255	0.008098656396051	4	0.032394625584204	6.4789
1	0	60087690	0.30774894304994	0	0
0	1	3162510	0.016197312792102	3	0.048591938376306	9.7184
0	0	120175380	0.61549788609987	0	0
		195249054	1		1	100

Quote: miplet

You also need to take into account the odds of splitting the jackpot, but that is beyond my math skills.

Jackpot amount times .5677 is allowance for multiple ticket holders claiming the prize. Expected after-tax return on a 1.00 "investment" in the lottery at 200,000,000 prize value is sixty-two cents.

miplet: Thanks for the analysis -- that is much like what I was looking for, and almost all of the other posts have focused exclusively on the jackpot. It is the biggest thing, but not the only thing. Right now, interest rates are such that the Powerball advertised (sum-of-annuity-payments) jackpot is 2.062 times the cash value, so it seems they would have to be advertising a jackpot of a little over $332 million for a positive expected value. (The MegaMillions multiplier is 1.629, but the figures for your table would be different for that lottery, too.)

I do have two questions, though:

First, with regard to your final comment, my crude thinking was that the expected value of a ticket would just be the ratio of the total funds expected to be paid out for all prizes in single drawing divided by the number of tickets in play for that drawing. If that is correct, the issue of splitting the jackpot between multiple winning tickets doesn't affect the expected value at all. Is that right?

Second, does your analysis consider the impact of a drawing with no jackpot winning ticket? Does it need to?

Quote: Doc

miplet: Thanks for the analysis -- that is much like what I was looking for, and almost all of the other posts have focused exclusively on the jackpot. It is the biggest thing, but not the only thing. Right now, interest rates are such that the Powerball advertised (sum-of-annuity-payments) jackpot is 2.062 times the cash value, so it seems they would have to be advertising a jackpot of a little over $332 million for a positive expected value. (The MegaMillions multiplier is 1.629, but the figures for your table would be different for that lottery, too.)

I do have two questions, though:

First, with regard to your final comment, my crude thinking was that the expected value of a ticket would just be the ratio of the total funds expected to be paid out for all prizes in single drawing divided by the number of tickets in play for that drawing. If that is correct, the issue of splitting the jackpot between multiple winning tickets doesn't affect the expected value at all. Is that right?

Second, does your analysis consider the impact of a drawing with no jackpot winning ticket? Does it need to?

My analisys was you buying 1 ticket and nobody else bought one. For example if you know that there will be 100,000,000 tickets sold you have to consider the posibliy of more than 1 jp winner. The final jp would have to be $205,908,251 for there to be a positive ev.
Here is a good link on how to analyse the lotto. It uses Powerball from February 15, 2006. The odds of winning have changed sence then, but it is a read.

Edit: I'm not going to correct all the typos in this post as it is way past my bedtime.

Quote: miplet

Here is a good link on how to analyse the lotto. It uses Powerball from February 15, 2006. The odds of winning have changed sence then, but it is a read.

Now that link is the kind of analysis I was looking for. Thanks very much!

To summarize, in that 2006 analysis performed at the request of a reporter, they considered a specific Powerball drawing with a jackpot having a cash value of $145.7 million and concluded that a $1 ticket in that drawing had an expected value of $0.9651. (Perhaps I should correct my terminology: when I have said a positive expected value, I have meant an expected value greater than the $1 ticket price.)

The analysis noted that there was no jackpot winner for that drawing, so the jackpot fund was even higher the next drawing. They did not analyze the expected value for a ticket in that next drawing or discuss any easy criteria for detecting when the expected value would exceed $1.

They did comment that the hard part of the analysis would be estimating the number of tickets for the drawing and that they obtained the lottery corporation's estimate from the reporter who had called them. In my initial post, I suggested that this figure could be obtained using the jackpot cash fund estimates for both the current and previous drawings.

The Powerball drawing earlier this week turned out to have a jackpot cash fund of $124.9 million, and there was a single winner. I think the fund would have had to roll over for another week to approximate the drawing that was analyzed as having a $0.9651 ticket value, and some additional time to give a ticket value greater than $1.

You would need to know how many tickets were sold during the previous drawings (and historical previous drawings of the same $$$ amount) to get a reasonable estimate of the total # of tickets sold for your drawing. With that information you can get an expected amount others should win.

This is my take on it... correct me if i'm wrong.

I.E. if 10 million tickets are going to be sold for the drawing and there are 40 million combinations assuming the numbers are randomly selected for other tickets (which they are for the most part) then there is a expected win of 25%. This is not to say that someone else will win 25% of the time, but that on average 12.5% of the winnings would go to someone else (1/4 you would split the wins (ON AVERAGE) and 3/4 times you would keep them all).

Now, it could happen where you split with more than one other person. Hell it is possible that it could split with 100 or 1000 people but the chances of that are so insanely remote as to border on impossible.

What this doesn't take into account though is that with a pool of buyers of tickets so large there are probably alot of people who buy 12345 or other distinct combinations 10, 20, 30, 40, 50 and if that hit you could easily be splitting it a few ways.

So you have to adjust the numbers and account for the lump sum payment (and income taxes - but you could probably write off the cost of the tickets purchased).

The biggest obstacle would be actually buying the tickets if the lottery didn't provide an easier method for buying bulk tickets. Perhaps if the "quick pick" feature made sure to not pick the same ticket twice for you but then again it would be impossible to walk into a 7-11 and buy 40 million quick picks - the roll of receipt paper for the ticket would not even come close to long enough.

Of course if you overcome all of the obstacles and find a time when it makes mathematical sense to implement this strategy and are willing to take the chance (however small) of having to split your winnings many ways then you should also have a family member open up a convenience store for you to buy your tickets at to also get a bonus.

My earlier suggestion was to use the increase in the estimated jackpot from one drawing to the next to indicate what the lottery operators were expecting as the number of tickets to be sold. They probably have better information and insight to that than an individual could hope to develop. Also, some of your comments seem to apply to that absurd situation where someone or some group tries to purchase every possible ticket combination. I have no interest in that scenario.

In regard to one of your comments, I think that if you are going to buy a lottery ticket with fantasies of hitting the jackpot, it would be the epitome of foolishness to choose 123456 or anything like that, knowing that if you did hit, you would have to split so many ways. Better to just let their computer choose a random group for you.

Quote: Doc

....
In regard to one of your comments, I think that if you are going to buy a lottery ticket with fantasies of hitting the jackpot, it would be the epitome of foolishness to choose 123456 or anything like that, knowing that if you did hit, you would have to split so many ways. Better to just let their computer choose a random group for you.

That's why I always choose 4 8 15 16 23 42.

Quote: Doc

In regard to one of your comments, I think that if you are going to buy a lottery ticket with fantasies of hitting the jackpot, it would be the epitome of foolishness to choose 123456 or anything like that, knowing that if you did hit, you would have to split so many ways. Better to just let their computer choose a random group for you.

Ah, ok. I just meant IF you buy every single ticket (somehow) and the number that wins is 12345 then you will probably have to split it many ways and end up losing money.

The problem with the lottery idea of buying 10 tickets (or even 10000) when you are getting a +EV is that it doesn't change your odds of winning. You still aren't going to win (most likely). The difference is if you did win, you would win more money. But again for most normal folk it doesn't really matter if you win 20 million or 100 million in the lottery and even if you found a good time to "invest" in it you would surely lose anyway unless you did something like buy all possible combinations. If you had the $$$ to buy all possible combinations then the payout would matter to you.

Quote: miplet

That's why I always choose 4 8 15 16 23 42.

I'll be careful not to choose those numbers -- better for both of us. ;-)

Quote: CFTCFT

The problem with the lottery idea of buying 10 tickets (or even 10000) when you are getting a +EV is that it doesn't change your odds of winning. You still aren't going to win (most likely). The difference is if you did win, you would win more money. But again for most normal folk it doesn't really matter if you win 20 million or 100 million in the lottery and even if you found a good time to "invest" in it you would surely lose anyway unless you did something like buy all possible combinations. If you had the $$$ to buy all possible combinations then the payout would matter to you.

I guess I didn't follow that. Buying tickets does increase your odds of winning -- from zero to something like one over the US national debt. And I don't know what you mean by "if you did win, you would win more money" unless you are referring to buying multiple tickets with the same combination so that you get the biggest share of all those who have the winning combination. That sounds like a really foolish way to select lottery numbers. And I disagree with your last sentence -- was there a typo? If I had enough money to buy all possible combinations, I doubt I would ever even bother to buy a lottery ticket; what would be the fantasy?

What I meant was comparing just playing the lottery on a normal game vs waiting for the jackpot to be high enough to give you a +EV on your ticket purchase. My basic point was that unless you are investing the time and money to buy every combination then it doesn't matter if you wait for +EV or not - you would still win 20 million instead of 50 or 100 million. If you were spending 40 million on tickets then it would definitely matter that the jackpot is 100 million not 20.