I asked myself, "What is the minimum estimated jackpot necessary to negate the "house advantage" in a lottery game with a jackpot rollover feature?" Here is what I came up with. Raw data courtesy of johnph77's web site http://www.johnph77.com/math/stlot.html

Powerball---$99.6 million

Wild Card 2---$1.8 million*

Hot Lotto---$8.3 million

Megamillions---$110.4 million

The Pick (AZ)---$3.6 million

Classic Lotto (CT)---$5.7 million

Lotto (DE)---$2.5 million

Cash (KS)---$3.9 million*

Tri-State Megabucks (ME, NH, VT)---$4.2 million

Lotto (MD)---$11.1 million*

Megabucks (MA)---$4.1 million

Mass Millions (MA)---$11.3 million

Northstar Cash (MN)---$121 thousand

Gopher 5 (MN)---$619 thousand

Cash (MT)---$322 thousand*

Pick 5 (NE)---$326 thousand

Nevada Numbers---$19.7 million

Roadrunner 5 (NM)---$128 thousand

Super Lotto Plus (OH)---$11.6 million

Wild Money (RI)---$233 thousand

Dakota Cash (SD)---$256 thousand

Lotto (WA)---11.2 million*

Quinto (WA)---$2 million

Megabucks (WI)---$11.9 million*

Badger 5 (WI)---$131 thousand

* denotes 2 games for $1

You are assuming a single winner.  The numbers are much higher if you recognize that the larger a jackpot is the greater the likelihood you will split the prize.

The expectation values vary with the conditions under which the lottery is played, specifically the speed with which the jackpot rises.  I have nonetheless tracked these using the statistics associated with the probable number of winners for both Powerball and Megamillions.  Generally the expectation value exceeds a dollar in the region where the jackpots are greater than $200 million dollars.

The expectation value for tonite's 121 million MM jackpot is 0.68. 

In spite of this relatively low value, I still have tickets. 

Good luck.

I wish we could win anyone lottery ASAP

I hope we could win any lottery ASAP

Quote: Originally posted by KyngeRycharde on February 10, 2004

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Powerball---$99.6 million

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I used the formula O * [1 - P] where O = odds of winning the jackpot and P = the total payout percentage minus the payout percentage for the jackpot (if it was given on johnph77's web site quoted above).

As prob987 correctly pointed out in his post, I went under the presumption that there would be a single winner. I also failed to mention that the amounts listed were also based on the size of the annuitized jackpot before taxes. Taking into consideration multiple winners, the cash option and taxes, the minimum jackpot necessary to negate the "house advantage" would, of course, be much higher.

Given the above conditions of a single winner, an annuitized jackpot and no taxes, the calculation for powerball would be 120,526,770*[1 - 0.1733] = 120,526,770 * 0.8267 = 99,639,480.76. This should give an expected value of 1.00.

I am not sure about how to factor in the above variables. Perhaps multiply $100 million by the inverse of the probability of two or more winners, then multiply by 2 (to take into consideration the cash option), and then multiply that by the inverse of [1 - 0.28] to take into consideration the taxes. Perhaps someone more learned and wiser in mathematical probability than I can clear up that question, as well as the question of how to deal with two games for $1 (denoted by an * above). Does one use the payout percentage as is (as I did) or divide the payout percentage by 2?

Regards and best wishes,

Rick

The following formula gives the expectation value for all the lower prizes of the MM.  The value is roughly 0.17.

=100000/2939677+5000/502195+100/12249+100/10685+7/261+7/697+4/124+3/70

You can see the general trends with this formula.  For the jackpot the situation is considerably more complex.  You need to use the Poisson distribution formula I posted in the Probability of a Rollover thread, sum the probabilities of winners times the cash value of the prize (before or after taxes depending on your outlook), dividing by the number of winners for each prize greater than one, and calculate the expectation value by adding this sum to 0.17.  The total expectation value will vary with the size of the rollover that produces it, but as a general trend, the expectation value seems to exceed 1 when the annuitized jackpot is over $220 million in either Powerball or MegaMillions.

For the current MegaMillions Jackpot, my calculation (based on my preferences for including issues like taxation) yields an expectation value of 0.89.

Mathematicians regard a wager as a "good bet" when the expectation value exceeds 1.  I don't know how many mathematicians buy tickets when the jackpot gets this high, though.  Like humans everywhere, some mathematicians are gamblers and some are not.  It depends, I guess, on how you look at improbable dvents.  The number of highly improbable dvents that take place in ordinary life (daily) thrill me, so I buy lottery tickets whenever the expectation value is not absurdly low.

Can math help crack lottery?NO

Leave that math to yourself

"In Search of a Fair Bet in the Lottery" @ http://www.williams.edu/Economics/wp/mathesonlottery.pdf examined 18,000 drawings in 34 American lotteries and finds approximately 1% of these drawings provided players with a fair bet, mostly in small states.  None of the popular lotteries like Mega Millions or Powerball has ever produced a fair bet since a large jackpot attracts a corresponding large number of buyers, diminishing the expected value of a ticket.

Thanks for the wonderful link lottobuddy.  I've downloaded the paper.   Eqn(4) is exactly the equation I use in calculating expectation values, with equ'n 3 being the form I posted on the other thread.  However, my results conflict with the statement that "there has never been a MB or PB drawing" that gave a fair bet."  I calculate that the expectation value for this drawing (217 million) is actually a fair bet with an expectation value of 1.02.  I have seen at least two or three other such drawings.   There are two possible reasons for the discrepancy. 

Although the paper is dated August 03, 2003, the data set of 18,000 drawings may have truncated much earlier, before the increase of odds to astromical proportions.  This increase (after which all expectation values >1 occurred), may have prdvented the authors from "catching them.) Another may have to do with the value of r as used in their calculation of the DV.  I actually do not estimate my values using this formula, relying instead on the running average of reported cash value/advertised value figures to calculate DV.

Historically jackpots over 150 million had greater sales than the recent one, probably because of a psychological jackpot saturation effect.  This has a tendency to increase rollovers actually, and the accumulation of funds in the jackpot that draw interest, and expectation values. 

Please post any similar finds you may have.  You are indeed a lottobuddy.

Interesting.  I will study this paper carefully when I have the time, and report any thing I find that can shed further light on the matter.

Lottobuddy, I too echo prob987's sentiments and wish to thank you for posting that link.

I have always enjoyed mathematics, and this will help me immensely in understanding what expected return is all about. Thank God I had teachers and professors who instilled in us a desire to wonder and ask questions, then attempt to find the answers.

LOL...perhaps I should have titled my original post "Expected Payout". And that brings to mind another question that I am curious about. What is the theoretical payout of a given lottery in it's history. For that, I would probably need to know the number of winners for each prize tier, the number of tickets sold, whether the cash option was chosen, and, perhaps, how much was withheld to fund non-lottery related projects. Just to satisfy my own curiosity.

Again, thank you both lottobuddy and prob987 for your contributions.

Regards and best wishes,

Rick

You're all welcome.  I've been trying to find out the expected payoffs for Canada's Lotto 6/49 and Lotto Super 7.  My hypothesis is that even though they offer tax-free cash jackpots, they have never had a positive expected return because of high sales.  I am requesting if prob987 or other statistical guru can plug the numbers into their complex formula and confirm this.

The last time the 6/49 jackpot exceeded the 1 in 13,983,816 odds was for a $15 million draw back in November 15, 1997.  Looking at that payout table, there was a total payout of $26,565,427.50, but the ticket sales of 38,085,190 is I think what lowers the expected payoff to well below the $1 ticket cost.

For Lotto Super 7(/47), you get 3 plays for $2 so the odds of winning the jackpot per $1 wager is 1 in 41,927,666.  The biggest jackpot ever in Canada was for $37.8 million in the May 17, 2002 draw.  The record sales of $102,092,406 was incredible for a nation of only 30 million people, and four winners split the jackpot.

Unlike Powerball, Canada's two national lotteries have never had a single draw where the total payouts exceeded sales.  For Powerball, the total sales is usually exceeded by the cash jackpot amount when it is dventually won.  For example, in the February 4 draw, Powerball sold only $17,795,366 worth of tickets, but paid out $50.9 million (including the $46.5 million cash jackpot).

LottoBuddy,

I have calculated the expected values of the Canadian Lotto 649 and Super 7 lotteries and no, they have never had a positive expected value. The reason is as you have surmised - high sales.

The highest expected value for the Lotto 649 lottery occured on the Sept 30, 2000 draw. The expected value was 79%. The jackpot was $15 million and the sales were 28,649,988 tickets. The expected value is usually around 70% when the jackpot is $10 million

The highest expected value for the Super 7 lottery occured just recently on Jan 30, 2004. The expected value was 70%. The jackpot was $30 million and the sales were 33,834,020 tickets.

By the way, my calculations show that the Mega Millions and Powerball lotteries never have had a positive expected value. I do factor in the taxes when calculating the expected value.

Good luck,
Jake

Well that may be the key to the discrepancy, Jake, and my calculation that the MegaMillions right now has a slightly favorable expectation value.   I don't factor in taxes.  People of course pay taxes on all kinds of income and usually report their earnings before taxes when asked "How much do you make?"  They do this on stock valuations as well.  So it seems to me that the calculation of a return on the ticket should be a before tax, rather than after tax calculation. 

It is worth noting that a favorable expectation value is a rarity for gamblers.  Gambling works because of a less than favorable expectation valuation, taxes or not.  Including a tax valuation reduces all gambling expectation values.

For the record, I will be more than happy to pay the taxes on any jackpot I win.  In an ideal world, taxes are simply the price you pay for living in a civilized world.

prob987,

I agree taxes are something we have to live with. But, nonetheless, I believe it is important that the tax factor is used to calculate the expected value. It is the after tax money that you truly "win". It is for the same reason that we use the cash value rather than the annutiy value to calculate expected value.

You are correct in stating that a positive expected value is a rarity. One lottery that currently does have a positve expected value is the Mass Millions with a jackpot of $48.5 million. With very low sales, one can reasonably expect to not have to share the jackpot if won. If I lived there, I would only buy tickets for this lottery to the exclusion of all other lotteries until the jackpot is won.

Good luck,
Jake

And you'd be smart to do it.

Quote: Originally posted by Jake649 on February 18, 2004

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The highest expected value for the Lotto 649 lottery occured on the Sept 30, 2000 draw. The expected value was 79%. The jackpot was $15 million and the sales were 28,649,988 tickets. The expected value is usually around 70% when the jackpot is $10 million

The highest expected value for the Super 7 lottery occured just recently on Jan 30, 2004. The expected value was 70%. The jackpot was $30 million and the sales were 33,834,020 tickets.

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Thanks for the replies.  For the 6/49 Sept. 30 draw, the prizes given out totaled $23,839,799.3, so with sales of $28,649.987, the actual return was 83%.

For Lotto Super 7, you get 3 plays for $2 so there were actually 101,522,391 "tickets" in play for January 30.  If we value the free plays as $2, the actual return for the $25 million bonus draw on January 23 was 84.6% ($46,517,297.30 / $54,964,354).  For Jan. 30, the potential return was 83% had the $30 million bonus jackpot been won ($56,317,199.5 / $67,681,594).

LottoBuddy,

I am using the term expected value. It is not calculated by simply dividing the amount of money paid out by the number of tickets sold. The reason is that the money paid out is based upon someone winning the jackpot, this is not a given. Expected value calculations does not depend upon the jackpot being won, only the probability that it is won and by how many tickets. In performing expected value calculations, you need to calculate the probability of no jackpot winner, 1 winner, 2 winners, etc. These are the calculations that prob987 has been posting.

I am sticking with my previous calculations.

Good luck,
Jake

I have reviewed lottobuddy's paper posted earlier.  As I surmised in my conversation with Jake, the issue comes down to taxation.  An expectation values you may have seen from me or will see from me do not include taxes.

Are you calculating for the government?I am surprised you guys are so fond of it..so whats the point?

Quote: Originally posted by goose on February 22, 2004

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Are you calculating for the government?I am surprised you guys are so fond of it..so whats the point?

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Looking for a fair bet is the best thing that gamblers like you can do.  The lotteries love players that stupidly buy tickets without even knowing the odds of winning, expected or potential value, and other important calculations.

Some mathematically-inclined people are interested in the math of lotteries.  To each his own.  I have seen a lottery newsgroup where all members do is post their useless picks; anytime a number from their "magic" wheels is drawn, they think it is evidence that the lottery balls are talking to them! 

Quote: Originally posted by Jake649 on February 19, 2004

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One lottery that currently does have a positve expected value is the Mass Millions with a jackpot of $48.5 million. With very low sales, one can reasonably expect to not have to share the jackpot if won. If I lived there, I would only buy tickets for this lottery to the exclusion of all other lotteries until the jackpot is won.

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LottoBuddy,

Here is my estimated expected value for the Mass Millions at the jackot level of $49.2 million.

Assuming an after-tax cash value of $16 million and 1 million tickets sold, I estimate the expected value of the jackpot prize itself at $1.10. The value of the secondary prizes bring the value higher. I did not bother calculating them because the value is already well over 1 dollar.

I would like to be part of a group to try to win the jackpot. You do not have to try to buy all the tickets to make it worthwhile. Even buying a couple of thousand tickets is a very "positive" gamble.

Good luck,
Jake

Quote: Originally posted by Jake649 on February 22, 2004

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Assuming an after-tax cash value of $16 million and 1 million tickets sold, I estimate the expected value of the jackpot prize itself at $1.10. The value of the secondary prizes bring the value higher. I did not bother calculating them because the value is already well over 1 dollar.

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Since nobody (including the research paper) has given an actual example showing the calculations, I'll give it a shot:

($2,657,320 / $13,983,816) + ($16,000,000 / 13,983,816) (P(0)/1 + P(1) /2 + P(2) /3)

= 0.19 + (1.144179814723 * 0.9650673333333)

= 0.19 + 1.10

= 1.29 expected value of Mass Millions

Please post any corrections.