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Probability of a MegaMillions rollover.

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

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New Jersey
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Posted: November 4, 2006, 8:03 am - IP Logged

The MM jackpot has rolled.  The cash value is $30.4M ; the advertised annuity is $52M.  Sixth round jackpots sell on average, 19.3M in tickets.

If an average amount is sold, the randomized probability of various numbers of winners will be as follows:

 

089.56%
19.88%
20.54%
30.02%

 

The long term probability model function is within 3% of actual events.  It suggests that the probability of the jackpot growing is as follows:

 

 

Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
$594,392,245.97$348,633,91444.89%0.29%
$518,095,732.34$303,883,07449.40%0.66%
$450,916,084.13$264,479,62653.74%1.33%
$391,763,903.80$229,784,59757.88%2.47%
$315 $339,679,968.69$199,235,36661.79%4.27%
$262 $293,819,676.11$172,336,54165.45%6.91%
$247 $253,439,347.10$148,651,92568.85%10.56%
$217 $217,884,166.82$127,797,44471.99%15.33%
$180 $186,577,565.91$109,434,91874.87%21.30%
$153 $159,011,870.77$93,266,57877.51%28.44%
$131 $134,740,070.99$79,030,23479.90%36.70%
$111 $113,368,570.60$66,495,02782.08%45.92%
$94 $94,550,805.43$55,457,68484.04%55.95%
$82 $77,981,623.11$45,739,22185.80%66.58%
$69 $63,392,334.73$37,182,04287.39%77.60%
$52 $50,546,357.72$29,647,38388.80%88.80%
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    California
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    Posted: November 4, 2006, 11:38 am - IP Logged

    Prob988...quick question if I may.  How did you calculate the figure for 0 winners, 89.56% in this instance?

    I tried taking the average number of tickets sold (19.3m) divided by total odds (175,711,536) and came up with a different number.  What factors should I be using?

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      Posted: November 4, 2006, 1:41 pm - IP Logged

      I use what is called a Poisson distribution, assuming that all purchases are effectively random.

      When the number of tickets purchased is very small compared to the odds, a Poisson distribution will give numbers that are close to the numbers obtained by dividing the number of tickets purchased by the probability.  However as the number of tickets purchased increases, there is a greater probability that each new purchaser will purchase a ticket with a set of numbers that has been already purchased by someone else. We all are intuitively aware that the larger the number of tickets that are sold, the more frequently we will see jackpots won by more than one person.  These possibilities are measured by Poisson calculations. 

      The mathematics of Poisson distributions are available on the internet and in many other places and I encourage you to investigate them.

      I have built a series of spreadsheets that do all of my calculations, and produce the results that I provide here from them.    I have arranged it so that I merely enter the sales for each drawing in a box on these spreadsheets and all of the values pop out.  I then cut and paste them here for informational purposes.

      Note that I only report values that have a probability of 1 hundredth of one percent.    The probability of there being six winners in the upcoming draw is not zero.    It is possible that there will be six winners, not very likely, but possible. 

      Sources of error in my calculation include the fact that purchases are not always random.  For instance people play their birthday numbers frequently, which tends to skew the probability of rollovers in someway by creating more doubly covered sets of numbers (those with all of the numbers being less than 31) and creating more uncovered sets  (those which do not represent birthdays.)  There are undoubtedly other such effects, however the Poisson calculations are a useful first approximation.

      The long term probabilities I post are based on a mathematical technique known as curve fitting.  I have graphed the sales figures for various jackpot sizes, made an estimate of the type of curve that fits, for MM an exponential function, and for PB a linear function that switches at a certain point to an exponential function, and have calculated certain constants.  These constants are automatically updated for each drawing by a series of linked spreadsheets.

      I hope you find them useful and I hope you find these comments useful. 

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        Posted: November 8, 2006, 4:00 am - IP Logged

        The MM jackpot has rolled.  The cash value is $37.1M ; the advertised annuity is $63M.  Seventh round jackpots sell on average, 21.7M in tickets.

        If an average amount is sold, the randomized probability of various numbers of winners will be as follows:

         

        088.39%
        110.90%
        20.67%
        30.03%

         

        The long term probability modelling function is pretty much on the money right now.  It predicts as follows:

         

        Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
        $592,026,467.62$348,637,80944.89%0.33%
        $516,018,875.83$303,877,78249.40%0.74%
        $449,095,861.20$264,467,56353.74%1.50%
        $390,171,617.16$229,767,73057.88%2.78%
        $315 $338,290,114.89$199,215,29061.79%4.81%
        $262 $292,609,592.02$172,314,53865.45%7.78%
        $247 $252,388,895.26$148,629,01668.85%11.89%
        $217 $216,975,455.44$127,774,43571.99%17.27%
        $180 $185,794,699.74$109,412,43474.88%23.99%
        $153 $158,340,729.41$93,245,09677.51%32.03%
        $131 $134,168,111.72$79,010,11079.91%41.33%
        $111 $112,884,652.93$66,476,51882.08%51.72%
        $94 $94,145,035.08$55,440,96584.04%63.01%
        $82 $77,645,213.28$45,724,40385.80%74.98%
        $69 $63,117,482.71$37,169,18487.39%87.39%
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          New Jersey
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          Posted: November 11, 2006, 1:24 am - IP Logged

          The MM jackpot has rolled.  The cash value is $44.3M ; the advertised annuity is $75M.  Eighth round jackpots sell on average, $23.2M in tickets.    If the current drawing results in average sales the probability of various numbers of winner is as follows:

           

          087.61%
          111.59%
          20.77%
          30.03%

           

          The long term model, a curve fit exponential function now overestimates the jackpot by 3%.  It is nonetheless useful for estimating the long term probability of the lottery going to higher jackpots.

          Here are the values:

           

          Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
          $590,231,123.34$348,629,85044.88%0.38%
          $514,437,385.11$303,861,01549.39%0.85%
          $447,705,186.67$264,444,53053.74%1.71%
          $388,951,173.30$229,740,49357.88%3.19%
          $315 $337,221,511.06$199,185,50661.79%5.50%
          $262 $291,676,401.83$172,283,52865.45%8.91%
          $247 $251,576,449.81$148,597,82368.85%13.61%
          $217 $216,270,657.82$127,743,86971.99%19.76%
          $180 $185,185,858.84$109,383,11474.88%27.45%
          $153 $157,817,410.96$93,217,48477.51%36.66%
          $131 $133,721,004.88$78,984,54079.91%47.30%
          $111 $112,505,450.80$66,453,22082.08%59.19%
          $94 $93,826,327.73$55,420,08484.04%72.11%
          $82 $77,380,392.00$45,706,01885.81%85.81%
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            Posted: November 15, 2006, 11:38 pm - IP Logged

            The MM jackpot has been won.    The cash prize is reset to $7.1M and the advertised annuity is $12M.  Average 1st draws produce 15.2M in sales.    Selling an average number of tickets would result in the following distribution for the probability of various numbers of winners:

             

            091.71%
            17.93%
            20.34%
            30.01%

             

            Here is the long term probability, which ignores minimums:

             

            Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
            $589,189,237.47$348,603,63244.87%0.20%
            $513,513,969.88$303,829,09949.39%0.44%
            $446,888,401.17$264,408,97153.73%0.88%
            $388,230,314.51$229,702,93657.88%1.64%
            $315 $336,586,910.96$199,147,25661.79%2.83%
            $262 $291,119,332.98$172,245,60565.45%4.58%
            $247 $251,089,038.57$148,561,01468.85%7.00%
            $217 $215,845,805.02$127,708,76872.00%10.17%
            $180 $184,817,167.09$109,350,15774.88%14.13%
            $153 $157,499,118.36$93,186,97877.52%18.87%
            $131 $133,447,924.48$78,956,68979.91%24.34%
            $111 $112,272,915.51$66,428,14282.09%30.46%
            $94 $93,630,140.14$55,397,83384.05%37.10%
            $82 $77,216,778.81$45,686,59485.81%44.15%
            $69 $62,766,224.91$37,136,68387.40%51.45%
            $52 $50,043,754.25$29,609,22188.82%58.87%
            $43 $38,842,712.35$22,981,93890.08%66.28%
            $33 $28,981,157.70$17,147,18591.22%73.57%
            $25 $20,298,906.46$12,010,18692.22%80.66%
            $16 $12,654,930.49$7,487,50193.12%87.46%
            $12 $5,925,066.66$3,505,66493.92%93.92%
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              Posted: November 18, 2006, 8:43 am - IP Logged

              MM has rolled over.  The cash jackpot is $9.4M.  The advertised annuity value is $16M.  Second round draws sell on average $14.9M tickets.  If this is the number of tickets that sell, the probability of various numbers of winners is distributed as follows:

               

              091.85%
              17.81%
              20.33%
              30.01%

               

               

              Lotteryreport has not yet posted the sales figures and I will update the long term jackpot evolution figures when she has done so.

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                Posted: November 19, 2006, 12:41 am - IP Logged

                The sales figures for the last MM drawing are now available.  Here is the long term probability model function data for the probability for the evolution of the jackpot:

                 

                 

                Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                $593,382,592.54$348,612,27344.87%0.21%
                $517,163,807.92$303,833,73749.38%0.46%
                $450,060,465.71$264,410,52453.73%0.94%
                $390,982,397.77$229,702,15957.88%1.75%
                $315 $338,969,815.43$199,144,76761.79%3.02%
                $262 $293,177,716.68$172,241,90965.45%4.88%
                $247 $252,862,158.13$148,556,51868.85%7.46%
                $217 $217,368,168.89$127,703,79972.00%10.83%
                $180 $186,119,109.77$109,344,97774.88%15.04%
                $153 $158,607,305.18$93,181,79277.52%20.09%
                $131 $134,385,795.39$78,951,65579.91%25.92%
                $111 $113,061,075.13$66,423,38282.09%32.43%
                $94 $94,286,700.69$55,393,43784.05%39.51%
                $82 $77,757,661.57$45,682,62685.81%47.01%
                $68 $63,205,425.22$37,133,18787.40%54.78%
                $56 $50,393,574.43$29,606,22588.82%62.68%
                $46 $39,113,966.49$22,979,45590.09%70.57%
                $34 $29,183,351.68$17,145,21991.22%78.34%
                $25 $20,440,396.15$12,008,73392.23%85.88%
                $16 $12,743,060.91$7,486,54893.12%93.12%
                $12 $5,966,294.22$3,505,19893.92%100.00%
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                  Posted: November 19, 2006, 12:51 am - IP Logged

                  The Powerball jackpot has rolled over.  The cash value is $26.1M; the advertised annuity is $54M.

                  On average, sixth drawings in a powerball jackpot run sell $16.89M in tickets. 

                  Were the average number of tickets to sell this time out, the randomized probability of various numbers of winners is as follows:

                   

                  089.08%
                  110.30%
                  20.60%
                  30.02%

                   

                  The model function for the long term evolution of jackpots suggests the following:

                   

                  Average Historical AnnuityCalculated AnnuityCalculated Cash Value ProjectionsSingle Draw Rollover ProbabilityOverall Probability from baseline
                  $563,775,073.13$272,491,28527.11%0.35%
                  $353 $444,264,170.36$214,727,68237.86%1.29%
                  $300 $355,325,804.62$171,740,80648.54%3.41%
                  $245 $289,139,099.33$139,750,56558.39%7.03%
                  $208 $239,883,866.00$115,943,86967.01%12.04%
                  $193 $203,228,800.60$98,227,25474.24%17.97%
                  $180 $175,950,606.43$85,042,79380.11%24.21%
                  $159 $155,650,550.50$75,231,09983.87%30.22%
                  $132 $139,545,043.85$67,446,77184.50%36.03%
                  $119 $124,128,268.44$59,995,33085.14%42.64%
                  $105 $109,400,224.25$52,876,77585.78%50.08%
                  $92 $95,360,911.30$46,091,10786.43%58.38%
                  $80 $82,010,329.57$39,638,32687.08%67.55%
                  $68 $69,348,479.07$33,518,43287.74%77.57%
                  $57 $57,375,359.80$27,731,42488.40%88.40%
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                    Posted: November 22, 2006, 8:21 am - IP Logged

                    MM has rolled over.  The cash jackpot is $13.6M.  The advertised annuity value is $23M.  Second round draws sell on average $15.9M tickets.  If this is the number of tickets that sell, the probability of various numbers of winners is distributed as follows:

                     

                    091.32%
                    18.29%
                    20.38%
                    30.01%

                     

                    The long term probability model suggests the following:

                     

                    Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                    $589,627,657.58$348,649,39844.86%0.22%
                    $513,867,975.02$303,852,36849.37%0.50%
                    $447,172,348.14$264,414,95453.72%1.01%
                    $388,456,333.32$229,695,91957.87%1.87%
                    $315 $336,765,232.15$199,130,74661.78%3.24%
                    $262 $291,258,568.45$172,222,45865.45%5.24%
                    $247 $251,196,422.46$148,533,53768.85%8.01%
                    $217 $215,927,400.07$127,678,81071.99%11.64%
                    $180 $184,878,041.39$109,319,19074.88%16.16%
                    $153 $157,543,496.54$93,156,15477.52%21.58%
                    $131 $133,479,316.98$78,926,90079.92%27.84%
                    $111 $112,294,228.99$66,400,06682.09%34.84%
                    $94 $93,643,771.71$55,371,96984.05%42.44%
                    $82 $77,224,696.31$45,663,29985.82%50.49%
                    $68 $62,770,035.34$37,116,19587.40%58.84%
                    $56 $50,044,761.93$29,591,68588.82%67.32%
                    $46 $38,841,968.32$22,967,42590.09%75.79%
                    $34 $28,979,501.67$17,135,70591.22%84.13%
                    $25 $20,297,002.31$12,001,70692.23%92.23%
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                      Posted: November 25, 2006, 5:14 am - IP Logged

                      MM has rolled over.  The cash jackpot is $18.4M.  The advertised annuity value is $31M.  Fourth round draws sell on average $16.8M tickets.  If this is the number of tickets that sell, the probability of various numbers of winners is distributed as follows:

                       

                      090.90%
                      18.67%
                      20.41%
                      30.01%

                       

                      The long term curve fitting model suggests the following for the probability for evolution of the jackpot.

                       

                      Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                      $587,406,671.23$348,654,28244.85%0.24%
                      $511,928,935.92$303,854,59449.37%0.54%
                      $445,482,042.33$264,415,14853.72%1.09%
                      $386,985,458.53$229,694,59557.87%2.03%
                      $315 $335,487,937.26$199,128,32461.78%3.51%
                      $262 $290,152,047.12$172,219,28065.44%5.68%
                      $247 $250,240,554.60$148,529,87868.85%8.69%
                      $217 $215,104,435.51$127,674,89171.99%12.62%
                      $180 $184,172,320.80$109,315,18474.88%17.52%
                      $153 $156,941,205.21$93,152,19977.52%23.40%
                      $131 $132,968,267.59$78,923,10179.92%30.19%
                      $111 $111,863,669.95$66,396,50182.09%37.78%
                      $94 $93,284,218.03$55,368,69784.05%46.02%
                      $82 $76,927,780.47$45,660,36085.82%54.75%
                      $68 $62,528,375.64$37,113,61787.40%63.80%
                      $56 $49,851,846.38$29,589,48388.82%72.99%
                      $46 $38,692,052.20$22,965,60590.09%82.18%
                      $34 $28,867,517.14$17,134,26891.22%91.22%

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                        Posted: November 26, 2006, 3:09 pm - IP Logged

                        Prob988, I hope MM continues to roll and get up to 365+ million.

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                          Posted: November 29, 2006, 11:21 pm - IP Logged

                          MM has rolled over.  The cash jackpot is $23.9M.  The advertised annuity value is $40M.  Fifth round draws sell on average $18.6M tickets.  If this is the number of tickets that sell, the probability of various numbers of winners is distributed as follows:

                           

                          089.94%
                          19.54%
                          20.51%
                          30.02%

                           

                          The long term curve fitting model suggests the following for the probability for evolution of the jackpot:

                           

                           

                          Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                          $583,563,750.38$348,679,34144.84%0.26%
                          $508,557,733.71$303,863,24649.36%0.59%
                          $442,529,465.60$264,411,35653.71%1.20%
                          $384,404,366.01$229,681,60957.86%2.23%
                          $315 $333,236,475.36$199,108,79461.78%3.85%
                          $262 $288,193,059.48$172,195,35365.44%6.23%
                          $247 $248,541,057.25$148,503,28268.85%9.53%
                          $217 $213,635,150.36$127,647,00271.99%13.83%
                          $180 $182,907,261.04$109,287,08874.88%19.22%
                          $153 $155,857,306.86$93,124,74177.52%25.66%
                          $131 $132,045,062.10$78,896,92579.92%33.10%
                          $111 $111,082,993.32$66,372,08982.09%41.42%
                          $94 $92,629,952.36$55,346,39784.05%50.46%
                          $82 $76,385,624.37$45,640,41185.82%60.03%
                          $68 $62,085,640.30$37,096,17087.41%69.95%
                          $56 $49,497,274.41$29,574,62188.83%80.03%
                          $46 $38,415,656.76$22,953,35590.09%90.09%
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                            Posted: December 3, 2006, 10:12 am - IP Logged

                            The MM jackpot has been won.    The cash prize is reset to $7.2M and the advertised annuity is $12M.  Average 1st draws produce 15.2M in sales.  Selling an average number of tickets would result in the following distribution for the probability of various numbers of winners:

                             

                            091.72%
                            17.93%
                            20.34%
                            30.01%

                             

                            Here is the long term probability, which ignores minimums, focusing on models generated from sales figures and what sales actually contribute to jackpots:

                             

                            Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                            $589,340,718.90$348,693,25944.83%0.19%
                            $513,574,074.30$303,864,66149.35%0.43%
                            $446,878,967.66$264,403,38953.70%0.88%
                            $388,169,264.02$229,666,81557.85%1.64%
                            $315 $336,488,871.33$199,089,24961.77%2.84%
                            $262 $290,996,170.42$172,172,73465.44%4.59%
                            $247 $250,950,309.16$148,478,93368.85%7.01%
                            $217 $215,699,137.59$127,621,99071.99%10.19%
                            $180 $184,668,587.67$109,262,24874.88%14.15%
                            $153 $157,353,324.52$93,100,71777.52%18.90%
                            $131 $133,308,516.95$78,874,20679.92%24.38%
                            $111 $112,142,593.43$66,351,03482.10%30.50%
                            $94 $93,510,865.27$55,327,26284.06%37.15%
                            $82 $77,109,913.33$45,623,36585.82%44.20%
                            $68 $62,672,646.84$37,081,31687.41%51.50%
                            $56 $49,963,953.81$29,562,00688.83%58.92%
                            $46 $38,776,872.20$22,942,98390.10%66.32%
                            $34 $28,929,219.51$17,116,45591.23%73.61%
                            $25 $20,260,625.97$11,987,53792.24%80.69%
                            $16 $12,629,922.85$7,472,70493.13%87.48%
                            $12 $5,912,843.58$3,498,43293.93%93.93%
                              Avatar
                              New Jersey
                              United States
                              Member #21206
                              September 4, 2005
                              949 Posts
                              Offline
                              Posted: December 6, 2006, 9:48 am - IP Logged

                              MM has rolled over.  The cash jackpot is $9.0M.  The advertised annuity value is $15M.  Second round draws sell on average $14.9M tickets.  If this is the number of tickets that sell, the probability of various numbers of winners is distributed as follows:

                               

                              091.87%
                              17.79%
                              20.33%
                              30.01%

                               

                              The long term modeling program I use, which does not account for jackpot minima, suggests the following:

                               

                              Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                              $581,306,498.72$348,783,89944.80%0.21%
                              $506,522,262.92$303,913,35849.32%0.46%
                              $440,699,603.43$264,419,76253.68%0.94%
                              $382,764,632.81$229,658,78057.84%1.75%
                              $315 $331,772,150.20$199,063,29061.76%3.02%
                              $262 $286,890,220.41$172,134,13265.43%4.89%
                              $247 $247,386,601.10$148,431,96168.84%7.47%
                              $217 $212,616,796.26$127,570,07871.99%10.85%
                              $180 $182,013,541.44$109,208,12574.88%15.08%
                              $153 $155,077,548.85$93,046,52977.52%20.14%
                              $131 $131,369,361.56$78,821,61779.93%25.97%
                              $111 $110,502,183.84$66,301,31082.10%32.50%
                              $94 $92,135,570.62$55,281,34284.06%39.58%
                              $82 $75,969,873.19$45,581,92485.83%47.08%
                              $68 $61,741,350.48$37,044,81087.42%54.86%
                              $56 $49,217,866.15$29,530,72088.84%62.75%
                              $46 $38,195,101.31$22,917,06190.11%70.64%
                              $34 $28,493,221.10$17,095,93391.24%78.39%
                              $25 $19,953,940.73$11,972,36492.25%85.92%
                              $16 $12,437,943.02$7,462,76693.14%93.14%
                                 
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