Doc
Doc
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April 23rd, 2010 at 9:54:19 AM permalink
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?
DJTeddyBear
DJTeddyBear
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April 23rd, 2010 at 9:59:14 AM permalink
Maybe you missed this article on the Wiz' main site, currently at the top of the "What's new" section:
http://wizardofodds.com/megabucks

There was also this thread:
http://wizardofvegas.com/forum/gambling/slots/1228-megabucks-odds/
In it, the Wiz talks about the above article, prior to posting it to the public.
I invented a few casino games. Info: http://www.DaveMillerGaming.com/ ————————————————————————————————————— Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
Doc
Doc
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April 23rd, 2010 at 10:14:51 AM permalink
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.
DJTeddyBear
DJTeddyBear
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April 23rd, 2010 at 11:22:19 AM permalink
Oh. Lottery, not slots.

Never mind.
I invented a few casino games. Info: http://www.DaveMillerGaming.com/ ————————————————————————————————————— Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
FleaStiff
FleaStiff
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April 23rd, 2010 at 5:25:37 PM permalink
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.
Doc
Doc
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April 23rd, 2010 at 6:53:01 PM permalink
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?
pacomartin
pacomartin
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April 23rd, 2010 at 9:35:15 PM permalink
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.
FleaStiff
FleaStiff
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April 24th, 2010 at 1:28:57 AM permalink
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!
miplet
miplet
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April 24th, 2010 at 2:11:33 AM permalink
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
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FleaStiff
FleaStiff
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April 24th, 2010 at 4:54:35 AM permalink
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.

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