New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 7, 2005, 12:22 pm - IP Logged

Mostly just to keep my mathematical muscles working, I did some more difficult work on exponential curve fitting and, playing around with the model, I came up with a function that matches the behavior of lottery pot growth pretty well, at least in the region in which we now are. The mathematical details of this approach can be found on the internet, http://mathworld.wolfram.com/LeastSquaresFittingExponential.html. As you can see the model is fairly complex computationally, but I had some other modelling spreadsheets available that I could modify appropriately.

To get this function to work well, I had to use only three data points to generate the curve fit, and I used data from the last large run, and not this run. Still, I think it is superior to my last model. The function that I was able to generate is: S = $561,000*exp(0.317*N) where S is the sales for the particular drawing and N is the number of the drawing, including the first drawing, which is N=1. The exp is the exponential function. (This function does not fit sales behavior in the early drawings very well, but those incorporating more data points, that fit early behavior better, do not model behavior of late drawings very well.)

Using this model I have constructed the following long range probabilities for rollovers from this point on. Here is the table for what I came up with:

Annuity Value Cash Value Single draw Prob of all

Rollover Prob. Rollovers

$733,420,458.18

$427,013,689

16.64%

0.36%

$561,321,837.59

$326,814,048

27.07%

2.18%

$435,918,446.74

$253,801,407

38.59%

8.05%

$344,540,547.36

$200,599,163

49.97%

20.85%

$277,956,059.98

$161,832,195

60.32%

41.73%

$229,437,829.75

$133,583,803

69.19%

69.19%

The last column is the probability that all the jackpot in question and all of the previous jackpots, including the one we have going on right now (note that the computed values are very close to the advertised values) will rollover. Thus we have a 40% chance of seeing a jack pot with a $200M cash value, and a 20% chance of seeing a jackpot with an annuity value of over $400M. The chances of larger jackpots are small, but not negligible.

I think it's a better model and it will be interesting to see if it works well, should we have more rollovers.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 7, 2005, 12:51 pm - IP Logged

Quote: Originally posted by fxsterling on November 7, 2005

looks good the roll overs seem right but still no luck for a 150 million roll

My model is just a model. Somehow I think that if a jackpot over $300M rolled - or maybe even before - my model would breakdown or at least become less accurate. Nothing about what I have written should be taken as absolute. It just gives one an idea of what the situation is approximately.

That said, I think it would take an extraordinary run to jump the jackpot by $150M. The record jump - I think - and Jake is better equipped than I to tell this, is the jump from $230 to $363 million in early May 2000 for the old big game. Players have become much more jaded since then.

Canada Member #2673 November 2, 2003 497 Posts Offline

Posted: November 7, 2005, 1:41 pm - IP Logged

Yes, the record jump is as you stated Prob988 - $133 million in May 2000.

Unfortunately we cannot know for certain if that jump was entirely based on sales or was it partially based on the poor estimate of the $230 million jackpot. I have noticed that Mega Millions (and Big Game) never adjusted their estimates after the draw to the actual value based upon sales. Therefore the $230 million jackpot may have grown to say $263 million by the time of the draw. Under this case, the jump was only $100 million.

Powerball on the other hand always states the actual jackpot value after the draw even if the jackpot was not won. A questinn for readers. Does anyone know of a web site that posts the actual cash value of the Powerball jackpot after the draw? There are a number of Powerball state web sites that posts the annuity value after the draw but I cannot find a site that posts the cash value. Thanks.

United States Member #2338 September 17, 2003 2063 Posts Offline

Posted: November 7, 2005, 2:52 pm - IP Logged

The cash value for the recent large Powerball was 164.4 million. It didn't go up from sales thanks to the idiotic cap.

While it would be nice if MM could finally break 300 the odds of it breaking 400 must be lower than that unless sales take off. At this point I can't see it lasting more than 3 rollovers.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 7, 2005, 3:08 pm - IP Logged

Quote: Originally posted by Jake649 on November 7, 2005

Yes, the record jump is as you stated Prob988 - $133 million in May 2000.

Unfortunately we cannot know for certain if that jump was entirely based on sales or was it partially based on the poor estimate of the $230 million jackpot. I have noticed that Mega Millions (and Big Game) never adjusted their estimates after the draw to the actual value based upon sales. Therefore the $230 million jackpot may have grown to say $263 million by the time of the draw. Under this case, the jump was only $100 million.

Powerball on the other hand always states the actual jackpot value after the draw even if the jackpot was not won. A questinn for readers. Does anyone know of a web site that posts the actual cash value of the Powerball jackpot after the draw? There are a number of Powerball state web sites that posts the annuity value after the draw but I cannot find a site that posts the cash value. Thanks.

The total was $565,015,173, with 34.61% represented by the last drawing. 36.6% of the final total jackpot (and I have used the announced jackpot of $363M, not the figure given in lottoreport - lottoreport only reports the "advertised" jackpots) came from the last jump. Therefore you are probably correct, the real jackpot jump was somewhat lower.

You can calculate cash value from the figures the lottoreport site, using another page on the lottoreport site: http://www.lottoreport.com/salescomparison.htm#pb2005. If you sum the sales for a drawing, and multiply by 0.318, you will get the cash value. The spreadsheet where I keep my records for megamillions automatically does it. For your reference, here are the cash values of the recent run of rollovers, as so determined from my spreadsheet.

$113,089,857.45

$97,939,964.29

$85,939,473.40

$74,631,212.53

$64,908,822.25

$54,573,259.95

$46,215,124.18

$38,822,651.15

$32,482,274.08

$25,807,824.52

$20,060,962.18

$14,420,071.19

$9,612,621.82

$4,664,967.33

The annuity values are determined by prevailing market conditions on the purchase of annuities. Since these can vary, the annuity jackpots are somewhat less indicative of the real state of affairs in terms of sales and the cash put into the prize by the lottery to make these purchases.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 9, 2005, 10:00 am - IP Logged

The Megamillions has rolled over again, and the cash value has risen. Based on sales, the cash value reached $131.5 and is expected to rise to $153.4, the annuity value from $165M to $192. This means that the lottery expects to sell about $70.2M tickets. From these values I estimate the probability of various numbers of winners as follows:

0

67.06%

1

26.79%

2

5.35%

3

0.71%

4

0.07%

5

0.01%

My long term prediction developed above was off by about 5%, which I don't consider all that bad, given that it is only a model. That model was developed using data from the last run. If I substitute the data from the last three draws on this run to generate the exponential function parameters I get the following equation: S = $2,416,000*exp(0.212*N) where N is the drawing number, including the starting draw, and S is the predicted sales for the draw. This function gives a long range probability outlook that runs like this:

Annuity Cash Value Prob of Rollover Prob of Rollover

$654,794,076.40

$381,235,662

30.50%

1.14%

$540,828,466.19

$314,882,351

38.28%

3.75%

$448,663,818.45

$261,222,045

46.00%

9.79%

$374,129,735.07

$217,826,646

53.36%

21.29%

$313,853,591.00

$182,732,535

60.17%

39.90%

$265,107,924.87

$154,351,725

66.31%

66.31%

Most likely the actual situation is somewhat between these values and the values from the equation derived in my previous post with different parameters for the exponential equation. The former equation, updated a little to reflect the data from this run with respect to existing sales, gives long range probabilities looking like this:

$729,669,651.32

$424,829,886

16.64%

0.52%

$557,571,030.73

$324,630,245

27.07%

3.15%

$432,167,639.89

$251,617,604

38.59%

11.63%

$340,789,740.51

$198,415,360

49.97%

30.14%

$274,205,253.13

$159,648,392

60.32%

60.32%

I suspect that the lower model, while somewhat further off the mark for the current drawing, may be better in the long run when, if there are many rollovers, a stronger case of lotto fever takes hold. Specifically I believe that there is a 30-40% of two more rollovers, which is quite signifcant under the circumstances, and that if this happens the annuity jackpot will be in the neighborhood of $400M annuity. I believe the next jackpot, if it occurs, will have a cash value greater than Jack Whitaker's prize. I also believe that any jackpot that is won, will still most probably have just one winning ticket.

Both equations give roughly a 10% chance of breaking $500M annuity, though the first equation takes longer to do it.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 12, 2005, 5:53 am - IP Logged

The Megamillions has rolled over again, and the cash value has risen. Based on sales, the cash value reached $131.5 and is expected to rise to $153.9, the annuity value from $262 to $310. This means that the lottery expects to sell about $90M tickets. From these values I estimate the probability of various numbers of winners as follows:

0

59.92%

1

30.69%

2

7.86%

3

1.34%

4

0.17%

5

0.02%

We now see that the most probable outcome, based on this data, is yet another rollover. The second most probable outcome, by far, is a single winner. The new jackpot will exceed Jack Whitaker's pot, and thus if a single winner is obtained, it will be a true record single winner jackpot on a cash basis, unlike the recent $340 annuity jackpot for powerball which was inflated via an alternate type of annuity.

The first long term probability equation given in my last post of this type, predicted within 1% the size of the current jackpot. This was pretty good, I think. The long term probabilities, given using the exponential model generated from the last three data points are as follows:

Annuity Value Cash Value Single draw All draw

Rollover Prop Rollover

$617,878,872.37

$363,751,272

34.32%

2.34%

$516,362,430.32

$303,987,560

41.50%

6.81%

$432,883,802.91

$254,842,884

48.52%

16.41%

$364,237,964.90

$214,430,415

55.17%

33.83%

$307,789,371.76

$181,198,582

61.32%

61.32%

Since this run and the previous run both now have run the same number of draws, 16, it is now possible to generate an exponential equation based on the average sales for both runs. Using 4 such data points, that model gives the following figures.

Annuity Value Cash Value Single draw All draw

Rollover Prop Rollover

$770,379,111.36

$448,531,838

19.06%

0.51%

$611,309,616.55

$355,918,043

27.83%

2.69%

$488,559,145.81

$284,449,992

37.27%

9.67%

$393,835,276.31

$229,299,650

46.69%

25.94%

$320,738,925.65

$186,741,330

55.56%

55.56%

Comparing the two model equations we see that, according to them, there is 16-26% chance of reaching an annuity jackpot of around $500M. There is a 7-10% chance of reaching $600M. The first model accomplishes these things in one more drawing.

BILOXI, MISSISSIPPI United States Member #19651 August 3, 2005 621 Posts Offline

Posted: November 12, 2005, 3:46 pm - IP Logged

hay could we take the cash put it in past $ VALUE and get an idea what jacks jack-pot was this one would be 375 ? or much less thanks {MY WIFE THINKS YOU DO REPORTS FOR WALLSTREET }

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 17, 2005, 9:16 am - IP Logged

The MegaMillions jackpot has been won. We can expect that the reset jackpot will sell around 14.5 million tickets. The probability of various numbers of winners is as is given in the following table:

0

92.34%

1

7.36%

2

0.29%

3

0.01%

Let's do a post mortem on the last $315M jackpot. The actual sales for this drawing was $104,813,365, giving a cash jackpot of $187.1M, an all time record. The final probabilities for numbers of winners is given in the following table.

0

55.07%

1

32.85%

2

9.80%

3

1.95%

4

0.29%

5

0.03%

Thus a rollover was more probable than a single winner, so having a single winner beat the odds, but not by all that much.

I have worked on a long term model for the growth of lottery sales using the average sales at each drawing point of the last two sets of rolls, that which went to $250M and that which went to $315M. The synthetic equation I generated is as follows:

S = $9,748,346.62 * exp(0.118N)

Here S is the sales for a particular drawing number, and N is the number of the drawing, including the first reset drawing which is 1. The results of this equation to not exactly match the behavoir of the lottery for every drawing, but I think it's not a bad approximation.

Here is a table comparing historical results and what this equation predicts:

$526,047,630.15

$306,276,620

51.28%

0.42%

$461,945,474.94

$268,954,921

55.25%

0.81%

$405,005,754.56

$235,803,350

59.04%

1.47%

$354,428,176.51

$206,355,961

62.62%

2.49%

$315

$309,501,868.77

$180,198,866

65.98%

3.98%

$250

$262

$269,595,388.37

$156,964,426

69.12%

6.03%

$200

$225

$234,147,846.42

$136,326,079

72.03%

8.72%

$172

$192

$202,661,024.79

$117,993,752

74.72%

12.10%

$150

$165

$174,692,373.60

$101,709,782

77.19%

16.20%

$131

$147

$149,848,791.14

$87,245,296

79.46%

20.98%

$111

$128

$127,781,098.82

$74,396,995

81.53%

26.41%

$97

$108

$108,179,133.39

$62,984,295

83.41%

32.39%

$84

$90

$90,767,387.60

$52,846,790

85.12%

38.84%

$72

$77

$75,301,137.94

$43,841,996

86.66%

45.63%

$61

$65

$61,563,004.97

$35,843,350

88.06%

52.65%

$51

$53

$49,359,898.12

$28,738,430

89.32%

59.79%

$41

$42

$38,520,301.70

$22,427,376

90.45%

66.94%

$32

$32

$28,891,864.29

$16,821,485

91.47%

74.00%

$23

$23

$20,339,257.41

$11,841,968

92.39%

80.90%

$15

$15

$12,742,273.46

$7,418,835

93.21%

87.57%

$12

$12

$5,994,136.20

$3,489,919

93.95%

93.95%

The first column shows the historical values for the jackpots for the $250M run, the second, the historical values for the jackpots for the $315M run. The third column shows what the equation predicts as an annuity value from the calculated cash value. The fourth column is the cash value calculated from the sales predicted by the equation given above. The fifth column gives the probability of a rollover for a jackpot having sales predicted by the equation. The fifth column contains the probability that that jackpot and all previous jackpots in the series will rollover. (The calculated sales figures are not shown themselves.)

One can see that for several regions - not including the earliest drawings where lottery policies inflate the prize - the equation represents a decent match with what actually occurs. This gives one a rough idea of how likely various jackpot sizes are at the time of a reset. It is just an approximation, of course, but it is a reasonable one I think. This fit is very good in the middle drawings, and not all that bad in the higher drawings.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 20, 2005, 11:39 pm - IP Logged

The Megamillions has rolled over again, and the cash value has risen. Based on sales, the cash value reached 5.4 million and is expected to rise to $9.4, the annuity value from $12 to $16M. This means that the lottery expects to sell about $12.6M tickets. From these values I estimate the probability of various numbers of winners as follows:

The probability of exceeding the large last prize, 315 M, is about 3.7%, of matching it, around 6.7%

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 20, 2005, 11:41 pm - IP Logged

The Megamillions has rolled over again, and the cash value has risen. Based on sales, the cash value reached 5.4 million and is expected to rise to $9.4, the annuity value from $12 to $16M. This means that the lottery expects to sell about $12.6M tickets. From these values I estimate the probability of various numbers of winners as follows:

0

93.08%

1

6.67%

2

0.24%

3

0.01%

The probability of exceeding the large last prize, 315 M, is about 3.7%, of matching it, around 6.7%