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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: August 5, 2006, 11:04 pm - IP Logged

The MM jackpot has rolled over.  The cash value is now $8.6M and the advertised annuity jackpot is $15M.

The average number of tickets sold on a second draw is 15.2M.  If this is the number of tickets sold, the distribution of probabilities for various numbers of winners is given as:

 

091.73%
17.92%
20.34%
30.01%

 

The long term probability for jackpot evolution as determined by an exponential curve fitted model is as follows:

 

$541,198,243.99$305,175,67649.27%0.45%
$471,058,240.59$265,624,50853.62%0.92%
(Average Annuity, M)$409,293,967.11$230,796,32057.76%1.71%
$315 $354,905,240.38$200,127,12261.67%2.96%
$262 $307,011,313.54$173,120,26865.34%4.81%
$247 $264,836,613.63$149,338,42468.74%7.36%
$217 $227,698,182.32$128,396,47571.89%10.70%
$186 $194,994,616.35$109,955,29874.78%14.89%
$158 $166,196,328.73$93,716,26377.42%19.91%
$135 $140,836,972.79$79,416,40479.82%25.71%
$115 $118,505,890.39$66,824,15582.00%32.21%
$96 $98,841,461.91$55,735,60283.97%39.29%
$84 $81,525,250.34$45,971,18385.74%46.79%
$71 $66,276,844.66$37,372,77687.33%54.57%
$57 $52,849,318.95$29,801,14488.75%62.49%
$45 $41,025,233.84$23,133,67490.03%70.41%
$35 $30,613,115.35$17,262,39691.16%78.21%
$26 $21,444,354.26$12,092,23392.18%85.79%
$17 $13,370,475.74$7,539,46393.08%93.08%

 

There seems to be some concern about the failure of MM to produce an advertised annuity jackpot of more than $100M since early spring.

Calculating from the model function, we see that there is (now) a 47% probability  of reaching a jackpot in that general area from this (second) draw.    It can be shown that there is a 99% probability that any second draw will produce such a second jackpot in, on average, 7.3 rounds of runs. 

To understand what this means properly, one should discard notions of the lottery thus being "due" to produce a large jackpot.    In probability, the occurrence of past events does not influence future events, as much as one might like to believe otherwise.  It means irrespective of previous events that 99 times out of 100, such a drawing will occur within 7 to 8 rounds of draws if we start from a second draw.   (The probability is somewhat lower if we start from a reset jackpot.)  However this is not the same as saying that it is a certainty, nor does the fact that we have gone since the end of March this year without such a jackpot have any effect on next two or three drawings.  There is a 99% probability of such a run within 7 to 8 draws from this drawing.

The laws of probability dictate that the MM could never again reach $100M and also that every series of future jackpot could end up with a jackpot being over $100M.  Either situation is highly improbable, however.

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    Posted: August 9, 2006, 8:53 am - IP Logged

    MM has rolled over.    The cash jackpot is $13.2M, the advertised annuity jackpot is $23.0M.    The average number of tickets sold on a third drawing in an MM rolling sequence is 15.97M tickets.  If this is the number of tickets sold, the probability distribution for numbers of winners will be the following:

     

    091.31%
    18.30%
    20.38%
    30.01%

     

    The long term probability for the evolution of the jackpot is as follows:

     

     

    $541,956,738.63$305,603,38348.99%0.48%
    $471,251,713.57$265,733,60553.39%0.98%
    (Average Annuity, M)$409,062,783.09$230,665,95857.58%1.84%
    $315 $354,364,222.82$199,822,04861.54%3.20%
    $262 $306,253,852.17$172,693,14465.24%5.19%
    $247 $263,938,154.06$148,831,79268.69%7.96%
    $217 $226,719,186.86$127,844,43071.87%11.59%
    $186 $193,983,072.82$109,384,89974.78%16.13%
    $158 $165,189,872.92$93,148,73477.45%21.57%
    $135 $139,864,681.32$78,868,14079.87%27.85%
    $115 $117,589,792.39$66,307,57782.06%34.87%
    $96 $97,997,811.22$55,259,87784.04%42.49%
    $84 $80,765,593.94$45,542,82185.82%50.56%
    $71 $65,608,917.86$36,996,14087.41%58.92%
    $57 $52,277,793.59$29,478,86788.84%67.40%
    $45 $40,552,341.83$22,867,01590.12%75.87%
    $35 $30,239,166.72$17,051,53091.25%84.19%
    $26 $21,168,166.05$11,936,49492.26%92.26%

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      Posted: August 12, 2006, 4:43 am - IP Logged

      Prob988, MM Jackpot for 8/15 has rolled to 31 million. Use my quote for Probability of a MM rollover.

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        Posted: August 13, 2006, 9:02 am - IP Logged

        MM has rolled over.    The cash jackpot is $17.6M, the advertised annuity jackpot is $31.0M.    The average number of tickets sold on a third drawing in an MM rolling sequence is 16.9M tickets.  If this is the number of tickets sold, the probability distribution for numbers of winners will be the following:

         

        090.82%
        18.74%
        20.42%
        30.01%

         

        The modelling function gives the following probability chart for jackpot evolution:

         

        Historical Average Annuity Value$612,654,881.75$351,610,62844.14%0.23%
        $533,034,435.85$305,915,41548.75%0.52%
        $463,078,004.40$265,766,50753.19%1.06%
        $401,612,607.61$230,490,71457.42%2.00%
        $315 $347,607,637.30$199,496,55761.42%3.48%
        $262 $300,157,576.45$172,264,34865.17%5.66%
        $247 $258,466,816.11$148,337,47768.64%8.68%
        $217 $221,836,315.18$127,314,75571.85%12.65%
        $186 $189,651,879.41$108,843,68774.79%17.61%
        $158 $161,373,863.01$92,614,56577.48%23.54%
        $135 $136,528,120.28$78,355,26979.91%30.38%
        $115 $114,698,055.42$65,826,71082.12%38.02%
        $96 $95,517,637.40$54,818,81884.11%46.30%
        $84 $78,665,262.57$45,147,02085.89%55.05%
        $71 $63,858,362.27$36,649,14787.49%64.09%
        $57 $50,848,664.90$29,182,71288.92%73.26%
        $45 $39,418,033.08$22,622,52390.20%82.38%
        $35 $29,374,806.09$16,858,58491.34%91.34%
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          Posted: August 16, 2006, 6:48 am - IP Logged

          The MM jackpot has rolled.  The cash jackpot is $22.9M.  The advertised annuity is $40M.  The average number of sales for a 5th drawing is 18.9M.

          If this is how many tickets sell, the randomized probability for various numbers of winners is:

           

          089.82%
          19.64%
          20.52%
          30.02%

           

          The modeling function for the jackpot evolution suggests the following:

           

          (Average Annuity, M)$613,563,034.34$352,131,82843.88%0.25%
          $533,377,293.50$306,112,18648.53%0.56%
          $462,992,790.93$265,717,60253.02%1.16%
          $401,211,504.77$230,260,51657.29%2.19%
          $315 $346,981,849.35$199,137,40961.33%3.82%
          $262 $299,380,776.14$171,818,53265.11%6.23%
          $247 $257,598,062.34$147,838,88868.61%9.56%
          $217 $220,922,520.09$126,790,31671.85%13.94%
          $186 $188,729,891.19$108,314,54674.81%19.40%
          $158 $160,472,221.59$92,097,10177.51%25.93%
          $135 $135,668,534.58$77,861,94279.96%33.45%
          $115 $113,896,644.04$65,366,77082.18%41.84%
          $96 $94,785,968.39$54,398,90484.17%50.91%
          $84 $78,011,222.84$44,771,65885.96%60.48%
          $71 $63,286,882.71$36,321,16787.57%70.36%
          $57 $50,362,323.50$28,903,59489.00%80.35%
          $45 $39,017,554.95$22,392,68490.28%90.28%

           

          If this function is accurate, we have an even shot of seeing a $100+ annuity jackpot.

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            Posted: August 20, 2006, 1:21 am - IP Logged

            The MM jackpot has rolled.  The cash jackpot is $28.2M.  The advertised annuity is $49M.  The average number of sales for a 5th drawing is 19.9M, but sales have been lower than average.

            If, nonetheless, this is how many tickets sell, the randomized probability for various numbers of winners is:

             

            089.30%
            110.11%
            20.57%
            30.02%

             

            The long term model for jackpot evolution suggests the following:

             

             

            Historical Average Advertised Annuity Jackpots$613,661,187.06$352,188,16043.85%0.27%
            $533,395,932.01$306,122,88348.50%0.62%
            $462,951,860.06$265,694,11152.99%1.28%
            $401,127,260.24$230,212,16757.27%2.42%
            $315 $346,867,461.83$199,071,76161.32%4.23%
            $262 $299,246,842.65$171,741,66665.10%6.90%
            $247 $257,453,038.82$147,755,65768.61%10.60%
            $217 $220,773,086.56$126,704,55471.85%15.45%
            $186 $188,581,259.65$108,229,24574.81%21.51%
            $158 $160,328,395.21$92,014,55777.52%28.75%
            $135 $135,532,525.38$77,783,88479.97%37.09%
            $115 $113,770,655.47$65,294,46382.19%46.37%
            $96 $94,671,548.03$54,333,23684.18%56.42%
            $84 $77,909,389.89$44,713,21585.98%67.03%
            $71 $63,198,234.07$36,270,29187.58%77.96%
            $56 $50,287,121.79$28,860,43589.01%89.01%
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              Posted: August 20, 2006, 1:21 am - IP Logged

              The MM jackpot has rolled.  The cash jackpot is $28.2M.  The advertised annuity is $49M.  The average number of sales for a 5th drawing is 19.9M, but sales have been lower than average.

              If, nonetheless, this is how many tickets sell, the randomized probability for various numbers of winners is:

               

              089.30%
              110.11%
              20.57%
              30.02%

               

              The long term model for jackpot evolution suggests the following:

               

               

              Historical Average Advertised Annuity Jackpots$613,661,187.06$352,188,16043.85%0.27%
              $533,395,932.01$306,122,88348.50%0.62%
              $462,951,860.06$265,694,11152.99%1.28%
              $401,127,260.24$230,212,16757.27%2.42%
              $315 $346,867,461.83$199,071,76161.32%4.23%
              $262 $299,246,842.65$171,741,66665.10%6.90%
              $247 $257,453,038.82$147,755,65768.61%10.60%
              $217 $220,773,086.56$126,704,55471.85%15.45%
              $186 $188,581,259.65$108,229,24574.81%21.51%
              $158 $160,328,395.21$92,014,55777.52%28.75%
              $135 $135,532,525.38$77,783,88479.97%37.09%
              $115 $113,770,655.47$65,294,46382.19%46.37%
              $96 $94,671,548.03$54,333,23684.18%56.42%
              $84 $77,909,389.89$44,713,21585.98%67.03%
              $71 $63,198,234.07$36,270,29187.58%77.96%
              $56 $50,287,121.79$28,860,43589.01%89.01%

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                Posted: August 23, 2006, 2:38 am - IP Logged

                Prob988, MM Jackpot for 8/25 has rolled to 60 million.

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                  Posted: August 23, 2006, 12:42 pm - IP Logged

                  The MM jackpot has rolled.    The cash prize is estimated to be $34.7M, the advertised annuity is $60M.  The average sales for a MM 7th draw in a series is $22.0M.  If this is the number of tickets that sell on this drawing, the randomized probability of a rollover is as follows:

                   

                  088.22%
                  111.06%
                  20.69%
                  30.03%

                   

                  The long term exponential modeling function suggests the following probability of jackpot evolution:

                   

                  Historical Average Advertised Annuity Jackpots$613,527,893.75$352,111,66143.95%0.31%
                  $533,481,072.90$306,171,74648.58%0.70%
                  $463,197,954.12$265,835,34853.06%1.44%
                  $401,487,610.95$230,418,97757.32%2.72%
                  $315 $347,304,380.02$199,322,51461.35%4.74%
                  $262 $299,730,142.66$172,019,03865.11%7.73%
                  $247 $257,958,767.61$148,045,90168.61%11.88%
                  $217 $221,282,451.39$126,996,88571.84%17.31%
                  $186 $189,079,724.77$108,515,32074.80%24.10%
                  $158 $160,804,922.14$92,288,04277.49%32.22%
                  $135 $135,978,935.42$78,040,08579.94%41.57%
                  $115 $114,181,095.67$65,530,02082.15%52.01%
                  $96 $95,042,045.01$54,545,86984.15%63.30%
                  $84 $78,237,477.92$44,901,50985.94%75.23%
                  $69 $63,482,646.03$36,433,51987.54%87.54%
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                    Posted: August 26, 2006, 12:59 pm - IP Logged

                    MM has rolled over.    The cash value prize is estimated at $40.5M; the advertised annuity value is $70M.

                    The average sales for an 8th round draw is $23.7M.  If this is the number of tickets sold on this draw, the randomized probability of various numbers of winners will be as follows:

                     

                    087.36%
                    111.81%
                    20.80%
                    30.04%

                     

                    The current run is lagging behind average runs at this level, possibly because it's been quite some time since MM has offered a jackpot this large. 

                    Nevertheless, the long term exponential model, for what it's worth, predicts the following for the jackpot evolution:

                     

                    Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                    $608,745,894.29$352,202,98243.78%0.35%
                    $528,979,618.02$306,052,49348.45%0.80%
                    $458,995,568.96$265,561,72252.95%1.65%
                    $397,594,092.64$230,036,58257.25%3.11%
                    $315 $343,722,655.58$198,868,10861.30%5.44%
                    $262 $296,457,802.94$171,522,01565.09%8.87%
                    $247 $254,989,328.78$147,529,54068.61%13.62%
                    $217 $218,606,387.69$126,479,41071.85%19.86%
                    $186 $186,685,309.60$108,010,78674.83%27.63%
                    $158 $158,678,908.90$91,807,08377.54%36.93%
                    $135 $134,107,104.72$77,590,53979.99%47.63%
                    $115 $112,548,691.44$65,117,45782.21%59.54%
                    $96 $93,634,118.48$54,174,02684.21%72.43%
                    $82 $77,039,155.50$44,572,65486.01%86.01%

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                      Posted: August 30, 2006, 3:45 am - IP Logged

                      Prob988, MM Jackpot for 9/1 has rolled to 83 million.

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                        Posted: August 30, 2006, 3:45 am - IP Logged

                        MM has rolled over.    The cash value prize is estimated at $48.1M; the advertised annuity value is $83M.

                        The average sales for an 9th round draw is $26.4M.  If this is the number of tickets sold on this draw, the randomized probability of various numbers of winners will be as follows:

                         

                        086.04%
                        112.94%
                        20.97%
                        30.05%

                         

                        Sales for this run are below average however.  For what it's worth, the exponential model based on average sales gives the following long term jackpot evolution, even if not accurate immediately:

                         

                        Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                        $608,590,812.78$352,113,25643.76%0.41%
                        $528,782,033.72$305,938,17748.44%0.93%
                        $458,770,343.04$265,431,41352.94%1.92%
                        $397,353,079.31$229,897,13957.24%3.62%
                        $315 $343,475,216.23$198,724,94761.30%6.33%
                        $262 $296,211,239.39$171,379,36065.10%10.32%
                        $247 $254,749,247.80$147,390,63668.62%15.86%
                        $217 $218,377,006.99$126,346,69771.86%23.11%
                        $186 $186,469,714.31$107,886,04974.84%32.16%
                        $158 $158,479,265.98$91,691,57577.55%42.97%
                        $135 $133,924,841.85$77,485,08780.01%55.41%
                        $115 $112,384,645.81$65,022,54582.23%69.26%
                        $96 $93,488,660.18$54,089,86884.23%84.23%
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                          Posted: September 2, 2006, 10:33 am - IP Logged

                          MM has rolled over.    The cash value prize is estimated at $55.4M; the advertised annuity value is $95M.

                          The average sales for an 10th round draw is $32.M.  If this is the number of tickets sold on this draw, the randomized probability of various numbers of winners will be as follows:

                           

                          083.19%
                          115.31%
                          21.41%
                          30.09%

                           

                           

                          The exponential modeling program shows the following jackpot evolution long term:

                           

                          Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                          $608,444,635.38$352,028,68243.76%0.41%
                          $528,631,731.95$305,851,21648.44%0.93%
                          $458,620,001.07$265,344,42952.95%1.92%
                          $397,205,840.58$229,811,95157.25%3.63%
                          $315 $343,333,452.89$198,642,92661.30%6.33%
                          $262 $296,076,694.31$171,301,51665.10%10.33%
                          $247 $254,623,153.32$147,317,68268.62%15.87%
                          $217 $218,260,184.04$126,279,10671.87%23.13%
                          $186 $186,362,654.82$107,824,10774.85%32.18%
                          $158 $158,382,201.34$91,635,41677.56%42.99%
                          $135 $133,837,799.44$77,434,72780.02%55.43%
                          $115 $112,307,495.61$64,977,90882.24%69.27%
                          $96 $93,421,153.01$54,050,81084.24%84.24%

                           

                          Actually the first value for the current drawing reflects a ninth drawing.    Expecting the lagging sales of MM may return to historical averages, now that the jackpot has reached higher values, I am treating this as a kind of reset.    Normally a tenth drawing would produce a jackpot over $100M

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                            Posted: September 6, 2006, 7:04 am - IP Logged

                            MM has rolled over.    The cash value prize is estimated at $66.5; the advertised annuity value is $114M.

                            Although this is an eleventh drawing coming up, it is more typical of an average 10th drawing, as sales have been lower for this run than on average.  Possibly the long run without an annuity jackpot over $100M played a roll.

                            A tenth round drawing sells on average (including this recent draw). sells 30.7M tickets.  If this is the number of tickets that sell, the randomized distribution of the probability of various numbers of winner is as follows:

                             

                            083.93%
                            114.70%
                            21.29%
                            30.08%

                             

                            The long term probability of the model program suggests the following behavior.

                             

                             

                            Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                            $602,264,881.84$351,321,18143.80%0.49%
                            $523,191,037.08$305,194,77248.48%1.11%
                            $453,838,608.00$264,739,18852.99%2.30%
                            $393,012,434.27$229,257,25357.30%4.33%
                            $315 $339,664,289.72$198,137,50261.36%7.56%
                            $262 $292,874,818.16$170,843,64465.15%12.32%
                            $247 $251,837,689.92$146,905,31968.68%18.92%
                            $217 $215,845,706.29$125,909,99571.92%27.54%
                            $186 $184,278,612.27$107,495,85774.90%38.29%
                            $158 $156,592,407.62$91,345,57177.61%51.13%
                            $131 $132,309,971.99$77,180,81780.06%65.88%
                            $111 $111,012,842.71$64,757,49282.28%82.28%
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                              Posted: September 9, 2006, 9:30 am - IP Logged

                              MM has rolled over.    The cash value prize is estimated at $78.8; the advertised annuity value is $135M.

                              Because of slow sales early in the drawing, I will treat the up coming drawing as an 11th round drawing rather than as a 12th round drawing.

                              An eleventh round drawing sells on average sells 34.8M tickets.  If this is the number of tickets that sell, the randomized distribution of the probability of various numbers of winner is as follows:

                               

                              082.01%
                              116.27%
                              21.61%
                              30.11%
                              40.01%

                               

                              Here is the model projections for long term evolution of the jackpot:

                               

                               

                              Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                              $602,006,566.48$351,393,46343.79%0.59%
                              $522,967,218.89$305,257,90348.47%1.35%
                              $453,644,956.44$264,794,24152.99%2.79%
                              $392,845,162.20$229,305,17657.29%5.26%
                              $315 $339,520,085.24$198,179,13161.35%9.19%
                              $262 $292,750,784.98$170,879,71765.15%14.97%
                              $247 $251,731,295.25$146,936,48968.67%22.98%
                              $217 $215,754,735.23$125,936,83871.92%33.47%
                              $186 $184,201,127.93$107,518,88174.89%46.53%
                              $158 $156,526,716.21$91,365,22477.60%62.13%
                              $131 $132,254,592.39$77,197,49580.06%80.06%
                                 
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