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# Probability of a MegaMillions rollover.

Topic closed. 352 replies. Last post 10 years ago by Prob988.

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United States
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June 2, 2005
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 Posted: March 3, 2006, 1:44 pm - IP Logged

My dad usually buys MM when it reaches \$250 million or more or if he calls me to know the winning numbers after he buys a ticket. I usually wait until \$300 million or more.

United States
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September 17, 2003
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 Posted: March 4, 2006, 12:26 am - IP Logged

What's the probability of a 400 or 500 million run not happening by now? I can only imagine how difficult it is calculating a run from a set jackpot level. Just looking for an educated guestamate.

New Jersey
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September 4, 2005
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 Posted: March 4, 2006, 7:21 am - IP Logged

The MM jackpot has rolled.  The cash value is now \$10.6M; the annuity value is now \$18M.    The average sales on the second drawing in a run is \$15.5M.  If this is the quantity actually sold, the randomized probability of a rollover/numbers of winners is as follows:

 0 91.56% 1 8.08% 2 0.36% 3 0.01%

The most updated exponential modeling function for the long term probabilities of jackpot evolution looks like this:

 \$437,079,382.81 \$260,260,905 54.64% 1.01% 19 Average Annuity \$380,371,059.99 \$226,493,677 58.67% 1.85% 18 \$315 \$330,339,354.46 \$196,702,070 62.48% 3.15% 17 \$256 \$286,198,187.27 \$170,418,012 66.03% 5.05% 16 \$231 \$247,254,029.36 \$147,228,536 69.34% 7.64% 15 \$192 \$212,895,005.08 \$126,769,298 72.39% 11.02% 14 \$158 \$182,581,278.63 \$108,718,852 75.20% 15.22% 13 \$141 \$155,836,572.38 \$92,793,595 77.77% 20.24% 12 \$120 \$132,240,683.79 \$78,743,316 80.10% 26.03% 11 \$102 \$111,422,883.31 \$66,347,262 82.22% 32.50% 10 \$87 \$93,056,089.66 \$55,410,672 84.14% 39.52% 9 \$75 \$76,851,730.83 \$45,761,712 85.87% 46.97% 8 \$62 \$62,555,210.16 \$37,248,784 87.42% 54.70% 7 \$51 \$49,941,906.21 \$29,738,135 88.82% 62.57% 6 \$41 \$38,813,643.59 \$23,111,761 90.07% 70.45% 5 \$32 \$28,995,579.32 \$17,265,550 91.18% 78.22% 4 \$23 \$20,333,455.73 \$12,107,649 92.18% 85.79% 3 \$15 \$12,691,176.86 \$7,557,019 93.07% 93.07% 2

The first column gives the average of long term annuity advertised jackpots since the matrix change, in millions.   The second column the calculated (from the model) value of the annuity (ignoring jackpot minimums for the first several drawings).    The third column gives calculated cash values.  The fourth column gives the percentage probability of the rollover of a particular draw.  The fifth column the probability of a particular drawing and all previous drawings rolling over.  The fifth column represents the drawing number in the sequence.

Obviously the program is not consistent, right now, with the jackpots because of two factors, the most obvious being the \$12M minimum in drawings, the second being the unusually high sales for the first drawing (close to \$20M) resulting from the large prize recently obtained.  Generally the model performs well in the intermediate level prizes.

Sparta, NJ
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July 9, 2005
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 Posted: March 4, 2006, 8:44 am - IP Logged

Validates my point - SAVE YOUR MONEY. Unless you're a college professor needing milk next Tuesday night.

|||::> *'`*:-.,_,.-:*''*:--->>> Chewie  <<<---.*''*:-.,_,.-:*''* <:::|||

I only trust myself - and that's a questionable choice

New Jersey
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September 4, 2005
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 Posted: March 5, 2006, 2:27 am - IP Logged

From the modeling equation, and the history of the MM, it is possible to estimate how many series of draws it will take before a MM run consists of 18 draws, which would probably reach a \$400M annuity jackpot.

It turns out that the average draw, is the 7.63th draw, meaning that if you take the average of all draw numbers since the matrix change, it is 7.63.  The modelling equation now gives the current sales, S, as, S = 9,827,428.44*exp(0.125*N) where N is the draw number.  From this equation we see that the average sales should be around \$25,550,000 which gives a rollover probability of 86.4%.

We raise 0.864 to the (18-7.63)th power to see that the probability of this number of rollovers is on average 0.221 or 22.1%.  Subtracting 0.221 from 1 we see that the probability of the jackpot not reaching 18 rolls is 0.779 or 77.9%.  Now we ask ourselves how many series of jackpots are required before there is only a 5% chance of not reaching a 18th roll.  The answer is ln(0.05)/ln(0.779) which is approximately 12 such series where the number of draws in the series has reached the average value.  Reaching the average value takes 91 draws if one does it 12 times, or almost a year of drawings.  Then one has to add the additional drawings to reach the magic 18 figure.

Thus an 18 roll drawing is only likely every couple of years.    It can happen any time of course, but it is at any given point unlikely.

Sparta, NJ
United States
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 Posted: March 5, 2006, 7:13 am - IP Logged

Since my financial plans account for \$40M cash - take home - I have another month before I need to invest.  By the way, you're model is impressive.  Way above anything I have the knowledge to create.

|||::> *'`*:-.,_,.-:*''*:--->>> Chewie  <<<---.*''*:-.,_,.-:*''* <:::|||

I only trust myself - and that's a questionable choice

New Jersey
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September 4, 2005
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 Posted: March 5, 2006, 7:25 pm - IP Logged

Since my financial plans account for \$40M cash - take home - I have another month before I need to invest.  By the way, you're model is impressive.  Way above anything I have the knowledge to create.

Thanks.

I think it's useful, within its limitations.  It took me some time to figure out how to do it, and I am surprised actually at the good fit, at least for MM.

I have not, by the way, sucessfully modeled PB yet.  This may be because the data set under the new matrix is too small.

EU
Estonia
Member #4217
April 1, 2004
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 Posted: March 7, 2006, 1:15 pm - IP Logged

Those results look pretty good. Your estimates just keep getting better for the sales numbers.

The evolution of the jackpot numbers do look good, but I recognized that because of the adhoc way I was messing around with this stuff, I made a mistake on the rollover probabilities.  (The jackpot evolution is unchanged.)  The probabilities of rollovers are actually somewhat higher than I wrote, because I shifted the cells in the excel sheet for one of the formulas involved.

Here is the corrected rollover probability with jackpots.

 \$747,369,796.04 \$445,024,742 23.83% 1.24% \$568,249,867.08 \$338,366,966 34.05% 5.22% \$433,676,591.39 \$258,234,698 44.51% 15.34% \$332,571,333.95 \$198,031,112 54.43% 34.47% \$256,610,687.02 \$152,800,000 63.32% 63.32%

Actually we see that we have a 5% chance, if this formula is correct, of reaching a jackpot of over \$700M, not a 2% chance.  We are about 1 in 3 for two more rollovers.

I hope this is not bad news to anyone.

Hi Prob988

Is it possible that you could send me this exel file? My e-mail is daimler@autonet.ee

Good Luck!

United States
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September 17, 2003
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 Posted: March 7, 2006, 4:15 pm - IP Logged

From the modeling equation, and the history of the MM, it is possible to estimate how many series of draws it will take before a MM run consists of 18 draws, which would probably reach a \$400M annuity jackpot.

It turns out that the average draw, is the 7.63th draw, meaning that if you take the average of all draw numbers since the matrix change, it is 7.63.  The modelling equation now gives the current sales, S, as, S = 9,827,428.44*exp(0.125*N) where N is the draw number.  From this equation we see that the average sales should be around \$25,550,000 which gives a rollover probability of 86.4%.

We raise 0.864 to the (18-7.63)th power to see that the probability of this number of rollovers is on average 0.221 or 22.1%.  Subtracting 0.221 from 1 we see that the probability of the jackpot not reaching 18 rolls is 0.779 or 77.9%.  Now we ask ourselves how many series of jackpots are required before there is only a 5% chance of not reaching a 18th roll.  The answer is ln(0.05)/ln(0.779) which is approximately 12 such series where the number of draws in the series has reached the average value.  Reaching the average value takes 91 draws if one does it 12 times, or almost a year of drawings.  Then one has to add the additional drawings to reach the magic 18 figure.

Thus an 18 roll drawing is only likely every couple of years.    It can happen any time of course, but it is at any given point unlikely.

Thanks for answering my question. I think Mega is long overdue for an eighteenth roll. While the new matrix has only been around for nine months Megamillions has been around for almost four years and has yet to roll once it passes the 300 million level. Reminds me of the string of low jackpots Powerball was plagued with a few years ago.

Morrison, IL
United States
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May 13, 2004
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 Posted: March 7, 2006, 4:20 pm - IP Logged

Powerball was plagued with a string of low jackpots just last year. Over a four month span, nine jackpots were awarded. In contrast, ever since the matrix change last August, only four Powerball jackpots were awarded.

New Jersey
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 Posted: March 8, 2006, 7:17 am - IP Logged

The MM jackpot has rolled.  The cash value is now \$18.1M; the annuity value is now \$31M.    The average sales on the third drawing in a run is \$15.2M.  If this is the quantity actually sold, the randomized probability of a rollover/numbers of winners is as follows:

 0 91.71% 1 7.93% 2 0.34% 3 0.01%

Lottoreport has not posted the sales yet, but it seems as if they were unusually large.  I will update the long term model when they come out.  Too many new records may have some impact on the accuracy of the model, but we will see.

New Jersey
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 Posted: March 8, 2006, 9:35 pm - IP Logged

The Megamillions has stepped into record territory, territory certainly not predicted by my exponential model, since the model works with average data.  However, I would not be surprised to see the model pick up as the run proceeds, assuming it does so.

With that in mind, I post the data, noting that right now it is considerably off the mark:

 \$437,034,541.61 \$260,234,204 54.72% 1.09% Average Annuity \$380,450,702.56 \$226,541,100 58.73% 1.99% \$315 \$330,510,011.28 \$196,803,689 62.52% 3.38% \$262 \$286,432,538.13 \$170,557,557 66.06% 5.41% \$247 \$247,529,920.07 \$147,392,816 69.36% 8.19% \$201 \$213,194,610.35 \$126,947,700 72.40% 11.81% \$169 \$182,890,390.42 \$108,902,914 75.20% 16.31% \$147 \$156,143,995.74 \$92,976,652 77.76% 21.69% \$135 \$132,537,724.72 \$78,920,191 80.09% 27.90% \$113 \$111,702,915.46 \$66,514,009 82.20% 34.84% \$92 \$93,314,188.32 \$55,564,358 84.12% 42.38% \$79 \$77,084,364.39 \$45,900,235 85.84% 50.38% \$67 \$62,759,980.60 \$37,370,716 87.40% 58.69% \$56 \$50,117,331.38 \$29,842,593 88.79% 67.15% \$45 \$38,958,974.99 \$23,198,299 90.04% 75.63% \$34 \$29,110,650.08 \$17,334,069 91.15% 84.00% \$26 \$20,418,554.21 \$12,158,321 92.15% 92.15%

Note that the model calculates cash values.  The better advertised annuity rates do not reflect the real value of prizes, since they vary with interest rates, a variable not addressed in thse calculations.  Even so, according the model, the run seems almost to have skipped a whole drawing.    When reality and mathematic models conflict, reality should always rule the day.

New Jersey
United States
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 Posted: March 11, 2006, 5:09 am - IP Logged

The MM jackpot has rolled.  The cash value is now \$24.5M; the annuity value is now \$31M.    The average sales on the fourth drawing in a run is \$17.2M.  If this is the quantity actually sold, the randomized probability of a rollover/numbers of winners is as follows:

 0 90.68% 1 8.88% 2 0.43% 3 0.01%

has not updated the sales figures yet.  When they do so, I will update the long term model.

New Jersey
United States
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 Posted: March 11, 2006, 8:19 am - IP Logged

Here is the long term probability model equation data:

 \$500,840,878.27 \$298,227,978 50.67% 0.60% 20 \$437,037,769.53 \$260,236,126 54.85% 1.18% 19 Average Annuity \$380,684,100.45 \$226,680,078 58.84% 2.15% 18 \$315 \$330,910,099.13 \$197,041,923 62.59% 3.66% 17 \$262 \$286,947,545.34 \$170,864,220 66.11% 5.85% 16 \$247 \$248,117,913.73 \$147,742,940 69.39% 8.85% 15 \$201 \$213,821,901.30 \$127,321,223 72.41% 12.76% 14 \$169 \$183,530,177.70 \$109,283,879 75.19% 17.62% 13 \$147 \$156,775,215.41 \$93,352,515 77.74% 23.43% 12 \$135 \$133,144,073.90 \$79,281,244 80.06% 30.14% 11 \$113 \$112,272,026.21 \$66,852,888 82.16% 37.64% 10 \$92 \$93,836,929.68 \$55,875,626 84.07% 45.81% 9 \$79 \$77,554,253.97 \$46,180,033 85.79% 54.50% 8 \$67 \$63,172,689.53 \$37,616,465 87.34% 63.52% 7 \$56 \$50,470,268.93 \$30,052,751 88.73% 72.73% 6 \$45 \$39,250,940.89 \$23,372,151 89.98% 81.97% 5 \$36 \$29,341,544.43 \$17,471,556 91.09% 91.09% 4

With the unusually fast growth of the jackpot, the model is currently off by almost 24%.  In most runs it gets closer by the fourth or fifth draw, but here is what this equation, again based on average jackpot performance, says, for what it's worth.

New Jersey
United States
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 Posted: March 11, 2006, 8:24 am - IP Logged

Note my error about the new annuity value in the post on current rollover probability.  The post should have read:

The MM jackpot has rolled.  The cash value is now \$24.5M; the annuity value is now \$42M.    The average sales on the fourth drawing in a run is \$17.2M.  If this is the quantity actually sold, the randomized probability of a rollover/numbers of winners is as follows:

 0 90.68% 1 8.88% 2 0.43% 3 0.01%

The error was transcriptional and the calculation is not affected.

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