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Probability of a MegaMillions rollover.

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

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Posted: January 21, 2007, 7:04 pm - IP Logged

MM has rolled over.    The jackpot is $19.2M cash.    The advertised annuity is $33M.    Third round sales have averaged 15.89M in sales.    If this is how many tickets sell, the randomized probability distribution for numbers of winners is as follows:

 

091.35%
18.26%
20.37%
30.01%

 

 

The long term evolution probabilities as modeled by least squares curve fitting to an exponential function suggest the following:

 

 

Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
$600,628,825.51$349,456,77144.66%0.24%
$523,213,202.69$304,414,95449.20%0.53%
$455,096,610.13$264,783,48253.58%1.08%
$395,162,064.45$229,912,47457.75%2.02%
$315 $342,426,752.38$199,230,11061.68%3.50%
$262 $296,025,914.48$172,233,25965.37%5.68%
$247 $255,198,664.70$148,479,22368.79%8.69%
$217 $219,275,513.31$127,578,48071.96%12.63%
$180 $187,667,388.52$109,188,29974.86%17.55%
$153 $159,855,976.82$93,007,11477.51%23.44%
$130 $135,385,223.63$78,769,58579.92%30.24%
$109 $113,853,854.85$66,242,24382.10%37.84%
$92 $94,908,796.72$55,219,66484.07%46.09%
$81 $78,239,386.09$45,521,09785.84%54.83%
$67 $63,572,276.13$36,987,50687.43%63.88%
$55 $50,666,953.97$29,478,95588.85%73.07%
$45 $39,311,796.71$22,872,31890.12%82.24%
$34 $29,320,601.24$17,059,25991.25%91.25%
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    Posted: January 24, 2007, 1:06 am - IP Logged

    MM has rolled over.  The cash jackpot is now $25.0M; the advertised annuity is $43M.

    On average, 5th round draws sell 18.4M tickets.

    If the sales are average on this drawing the randomized probability of various numbers of winners is as follows:

     

    090.05%
    19.44%
    20.49%
    30.02%

     

    The long term model suggests the following evolution of the jackpot:

     

    Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
    $601,086,501.48$349,468,89644.65%0.26%
    $523,601,041.08$304,419,21049.19%0.58%
    $455,424,647.90$264,781,77253.57%1.19%
    $395,438,934.94$229,906,35857.74%2.22%
    $315 $342,659,877.68$199,220,85961.68%3.84%
    $262 $296,221,671.84$172,221,90265.37%6.22%
    $247 $255,362,530.49$148,466,58768.79%9.52%
    $217 $219,412,187.40$127,565,22571.96%13.84%
    $180 $187,780,901.84$109,174,94374.86%19.23%
    $153 $159,949,784.24$92,994,06177.51%25.69%
    $130 $135,462,284.17$78,757,14279.92%33.15%
    $109 $113,916,701.00$66,230,64082.10%41.47%
    $92 $94,959,594.18$55,209,06684.07%50.52%
    $81 $78,279,985.37$45,511,61985.84%60.09%
    $67 $63,604,257.02$36,979,21987.43%70.01%
    $55 $50,691,663.85$29,471,89888.85%80.07%
    $45 $39,330,383.64$22,866,50290.12%90.12%
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      Posted: January 31, 2007, 7:57 am - IP Logged

      MM has rolled over.  The cash prize is $37.5M, the advertised annuity is $65M.  Seventh round drawings in MM sell on average 21.1M tickets.  This suggests that the probability of various numbers of winners is as follows:

       

      088.67%
      110.66%
      20.64%
      30.03%

       

       

      The long term model for jackpot evolution is as follows:

       

      Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
      $605,760,751.90$349,477,35744.64%0.33%
      $527,653,223.27$304,415,32149.19%0.73%
      $458,932,462.64$264,768,72853.57%1.48%
      $398,470,391.63$229,886,76457.74%2.77%
      $315 $345,274,501.53$199,196,82861.68%4.79%
      $262 $298,471,560.84$172,195,13165.37%7.77%
      $247 $257,293,280.84$148,438,43168.79%11.89%
      $217 $221,063,703.83$127,536,75271.96%17.29%
      $180 $189,188,107.03$109,146,98574.86%24.02%
      $153 $161,143,239.96$92,967,25477.51%32.09%
      $130 $136,468,735.07$78,731,96379.92%41.40%
      $109 $114,759,550.64$66,207,43382.10%51.80%
      $92 $95,659,321.86$55,188,07084.07%63.10%
      $81 $78,854,510.94$45,492,98785.84%75.05%
      $67 $64,069,260.28$36,963,03587.43%87.43%
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        Posted: February 3, 2007, 1:17 pm - IP Logged

        MM has rolled over.  The cash prize is $44.7M, the advertised annuity is $7M.  Eight round drawings in MM sell on average 23.1M tickets.  This suggests that the probability of various numbers of winners is as follows:

         

        087.67%
        111.54%
        20.76%
        30.03%

         

        The long term model suggests the following:

         

        Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
        $602,015,183.19$349,481,54144.65%0.37%
        $524,400,166.86$304,424,51249.19%0.83%
        $456,111,277.88$264,781,48253.57%1.70%
        $396,027,899.96$229,901,91157.74%3.17%
        $315 $343,164,068.73$199,213,42761.68%5.48%
        $262 $296,652,292.11$172,212,43565.37%8.89%
        $247 $255,729,314.84$148,455,84968.79%13.60%
        $217 $219,723,593.47$127,553,82671.96%19.77%
        $180 $188,044,276.43$109,163,36674.86%27.47%
        $153 $160,171,508.17$92,982,68177.51%36.70%
        $130 $135,647,898.33$78,746,24779.92%47.35%
        $109 $114,071,016.07$66,220,44782.10%59.25%
        $92 $95,086,786.12$55,199,73284.07%72.16%
        $81 $78,383,678.50$45,503,25285.84%85.84%
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          Posted: February 7, 2007, 12:27 pm - IP Logged

          The MM jackpot has rolled over.  The cash jackpot is now $52.8M.    The jackpot's advertised annuity value is $91M.

          The average sales for a ninth drawing are 25.6M.

          The randomized probability of various numbers of winners for this amount of sales is as follows:

           

          086.43%
          112.61%
          20.92%
          30.04%

           

          The expectation value is 0.47.

          The long term probability rollover model gives the following projection:

           

          Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
          $602,273,754.17$349,451,14544.64%0.43%
          $524,606,774.74$304,387,22849.19%0.97%
          $456,274,997.43$264,739,77953.57%1.98%
          $396,156,375.28$229,857,76557.74%3.69%
          $315 $343,263,726.03$199,168,40461.68%6.39%
          $262 $296,728,522.06$172,167,75865.37%10.36%
          $247 $255,786,628.63$148,412,46168.79%15.85%
          $217 $219,765,756.38$127,512,43971.96%23.04%
          $180 $188,074,421.92$109,124,50074.86%32.01%
          $153 $160,192,235.38$92,946,70477.51%42.76%
          $130 $135,661,355.25$78,713,40279.92%55.17%
          $109 $114,078,970.44$66,190,87582.10%69.03%
          $92 $95,090,685.79$55,173,49784.07%84.07%
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            Posted: February 10, 2007, 11:09 am - IP Logged

            The MM jackpot has rolled over.  The cash jackpot is now $61.9M.  The advertised annuity is $106M.

            Tenth round draws sell on average $31.5M in sales.  If this drawing produces average sales, the probability of various numbers of winners resulting is as follows:

             

            083.57%
            115.00%
            21.35%
            30.08%

             

            The long term exponential form modelling equation for MM suggests the following:

             

            Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
            $598,413,915.98$349,451,14544.64%0.52%
            $521,244,686.90$304,387,22849.19%1.16%
            $453,350,832.72$264,739,77953.57%2.35%
            $393,617,497.41$229,857,76557.74%4.39%
            $315 $341,063,825.36$199,168,40461.68%7.60%
            $262 $294,826,855.13$172,167,75865.37%12.32%
            $247 $254,147,349.17$148,412,46168.79%18.85%
            $217 $218,357,326.65$127,512,43971.96%27.40%
            $180 $186,869,094.89$109,124,50074.86%38.08%
            $153 $159,165,599.06$92,946,70477.51%50.86%
            $130 $134,791,931.88$78,713,40279.92%65.62%
            $109 $113,347,863.76$66,190,87582.10%82.10%
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              Posted: February 14, 2007, 11:02 pm - IP Logged

              The MM jackpot has rolled over.  The cash jackpot is now $75.6M.  The advertised annuity jackpot is $130M.  The average number of tickets sold on an 11th round draw is 35.6M.  If this is the number of tickets that are sold, the randomized probability of various numbers of winners is:

               

              081.65%
              116.55%
              21.68%
              30.11%
              40.01%

               

               

              Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
              $600,936,136.50$349,467,47644.64%0.63%
              $523,443,168.26$304,402,33549.19%1.41%
              $455,264,256.27$264,753,67553.56%2.86%
              $395,279,922.92$229,870,47857.74%5.34%
              $315 $342,505,243.15$199,179,97261.68%9.26%
              $262 $296,073,672.28$172,178,22865.37%15.01%
              $247 $255,222,817.61$148,421,88568.79%22.96%
              $217 $219,281,920.15$127,520,87071.96%33.37%
              $180 $187,660,840.98$109,131,99774.86%46.37%
              $153 $159,840,371.29$92,953,32477.51%61.95%
              $130 $135,363,707.19$78,719,20279.92%79.92%
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                Posted: February 18, 2007, 11:37 am - IP Logged

                The MM jackpot has rolled over.  The cash value is now $88.3M.  The advertised annuity is $150M.  The averages sales for 12th round drawings is 41.6M.  Sales at this level would produce the following probabilities for various numbers of winner:

                 

                078.90%
                118.70%
                22.22%
                30.17%
                40.01%

                 

                 

                The long range model suggests the following:

                 

                Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                $593,925,788.07$349,624,31444.63%0.78%
                $517,337,520.15$304,539,35449.17%1.76%
                $449,954,465.69$264,873,19553.55%3.58%
                $390,670,241.74$229,974,54957.72%6.68%
                $315 $338,511,442.50$199,270,40261.66%11.57%
                $262 $292,621,656.70$172,256,61565.35%18.76%
                $247 $252,247,405.92$148,489,64068.78%28.71%
                $217 $216,725,773.01$127,579,23871.95%41.74%
                $180 $185,473,517.42$109,182,07774.85%58.01%
                $153 $157,977,498.80$92,996,08877.50%77.50%
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                  Posted: February 24, 2007, 5:35 pm - IP Logged

                  The MM jackpot has rolled over.  The cash jackpot is is 126.9M.  The advertised annuity is $216M.  Fourteenth draws sell on average 61.7M tickets.  This suggests the following probability for various numbers of winners:

                   

                  070.78%
                  124.46%
                  24.23%
                  30.49%
                  40.04%

                   

                   

                  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
                  $597,412,900.92$350,980,07944.43%1.33%
                  $520,254,674.76$305,649,62148.99%2.98%
                  $452,388,469.16$265,778,22653.39%6.09%
                  $392,695,263.51$230,708,46757.58%11.41%
                  $315 $340,190,798.75$199,862,09461.53%19.82%
                  $262 $294,009,348.27$172,730,49265.24%32.20%
                  $247 $253,389,443.32$148,866,29868.68%49.36%
                  $217 $217,661,317.40$127,876,02471.86%71.86%
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                    Posted: February 25, 2007, 12:53 pm - IP Logged

                    I always look forward to your predictions.

                    Just curious if you would do a spread as the jackpots get higher. A low-high estimates for a spread. I think the jackpot would go 217-265(or 270)-315 assuming it rolls at all. Sales seem very high (on level with pre-matrix 2004 levels) as the game grows due to the dearth of large jackpots recently.

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                      Posted: February 26, 2007, 10:37 pm - IP Logged

                      I always look forward to your predictions.

                      Just curious if you would do a spread as the jackpots get higher. A low-high estimates for a spread. I think the jackpot would go 217-265(or 270)-315 assuming it rolls at all. Sales seem very high (on level with pre-matrix 2004 levels) as the game grows due to the dearth of large jackpots recently.

                      I expect that if the jackpot continues to roll, my model will be less predictive in terms of the actual jackpot.  In fact, hidden in these tables is the fact that I am not actually predicting jackpots so much as sales.   

                      The higher the jackpot goes, of course, the fewer instances there are of previous sales and the weaker the model is at approximating it.  There are many factors at different times, jackpot fatigue, competing jackpots, economic conditions, even weather.

                      Graphically the exponential model I use breaks down as sales grow higher.  In the past I've used other cruder models that are based on the last 5 drawings rather than the entire set of drawings.

                      My general impression is that while the pure exponential model of MM has worked in the past, more recently the behavior of the MM system has become more like PB, which is grows linearly in the early draws and exponentially in the later draws.

                      I think another rollover will almost certainly give a jackpot higher than $250 million, and maybe higher than $270M.  It's hard to know, of course.    It is certainly the case that the probability of another rollover is favorable, but of course, it is less likely any previous drawing.  Two rollovers will certainly bring the jackpot to the mid/low 300's.

                      I expect that we will someday see a jackpot of over $400M, but even now with the jackpot over $200M, this still is unlikely for this run.  It's always going to be a long shot, but it seems likely that it will happen simply because there are so many repeats of the situation. 

                      It is interesting that now the winner of such a jackpot would be able to hide himself or herself relatively easily.    Such jackpots don't excite as much as they used to do.    This of course, leads to less drawings where the field is nearly saturated.

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                        Posted: February 27, 2007, 2:12 am - IP Logged


                        Even the people running the lottery have trouble estimating how many tickets will be sold at these levels. For the levels that are seen commonly they've got good data to work with and a 10% error is insignificant in real dollars.  That make sit pretty easy for them to make a relaible estimate When advertised jackpots go over $200 million  there's not much data to work with and there's a lot more unpredictability. For a $50 million jackpot there's a small but dependable minority who will spend $25, $50 or $100 on tickets. Above $200 million a lot of people act irrationally and spend amounts that are wildly out of proportion to what they would spend on smaller jackpots, and plenty of people who don't play at all for more typical jackpots will buy $50, $100 and even $500 worth of tickets

                        We're at a level where succeeding jackpots increase by ever bigger amounts. Since the last increase was $39 million sales would have to fall well short of estimates for the next jackpot to be much less than $270 if it rolls. For jackpots over $200 million it hasn't been unusual for the advertised value to be adjusted, and I don't recall ever seeing a final value that wasnt a bit higher than the estimate. I think MM and PB play it safe by lowballing their estimates, which is certainly safer than coming up several million shy of what they advertised.  Perhaps one of the biggest mysteries is how the supposed jackpot fatigue will affect sales. Some people may wait and hope it rolls past 300, but since this is the first time it's been this high since last April, perhaps a lot of people will be jumping at the chance. The last two rolls have seen bigger jumps than similar levels did last February and April, so perhaps that is an indication that  people are already reacting to the higher jackpot. If it does roll again I'd expect them to advertise it at 265 and finish closer to 270 or even 275. I think the low to mid 200's has always been a problematic level for estimates. It's high enough to be a bit unpredictable, but not quite high enough for a lot of people to get crazy. If it does roll to about 270 and rolls again we'll see  a good case of jackpot fever and I'd be very surprised if the next one doesn't  get  to about 340.

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                          Posted: February 27, 2007, 9:51 pm - IP Logged


                          Even the people running the lottery have trouble estimating how many tickets will be sold at these levels. For the levels that are seen commonly they've got good data to work with and a 10% error is insignificant in real dollars.  That make sit pretty easy for them to make a relaible estimate When advertised jackpots go over $200 million  there's not much data to work with and there's a lot more unpredictability. For a $50 million jackpot there's a small but dependable minority who will spend $25, $50 or $100 on tickets. Above $200 million a lot of people act irrationally and spend amounts that are wildly out of proportion to what they would spend on smaller jackpots, and plenty of people who don't play at all for more typical jackpots will buy $50, $100 and even $500 worth of tickets

                          We're at a level where succeeding jackpots increase by ever bigger amounts. Since the last increase was $39 million sales would have to fall well short of estimates for the next jackpot to be much less than $270 if it rolls. For jackpots over $200 million it hasn't been unusual for the advertised value to be adjusted, and I don't recall ever seeing a final value that wasnt a bit higher than the estimate. I think MM and PB play it safe by lowballing their estimates, which is certainly safer than coming up several million shy of what they advertised.  Perhaps one of the biggest mysteries is how the supposed jackpot fatigue will affect sales. Some people may wait and hope it rolls past 300, but since this is the first time it's been this high since last April, perhaps a lot of people will be jumping at the chance. The last two rolls have seen bigger jumps than similar levels did last February and April, so perhaps that is an indication that  people are already reacting to the higher jackpot. If it does roll again I'd expect them to advertise it at 265 and finish closer to 270 or even 275. I think the low to mid 200's has always been a problematic level for estimates. It's high enough to be a bit unpredictable, but not quite high enough for a lot of people to get crazy. If it does roll to about 270 and rolls again we'll see  a good case of jackpot fever and I'd be very surprised if the next one doesn't  get  to about 340.

                          There are only three examples of 15th draws, two examples of 16th draws, and one example of 17th draws in MM.  In 14th draws (4 examples) the numbers have been all over the place, ranging from 48M to 76M.    One 15th draw had sales of 96M, another of 58M.

                          I don't have any idea what would happen with a rollover and I think you're right on as usual - I don't think any one knows.

                          There is still some probability of a rollover, so maybe we'll find out. 


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                            Posted: February 27, 2007, 9:55 pm - IP Logged

                            Prob988, I hope MM continues to roll until it reaches between 315 and 365 million and one ticket hits the jackpot.


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                              Posted: February 28, 2007, 1:47 am - IP Logged

                              Prob988, MM Jackpot has rolled to 267 million for 3/2. Lets see how the probability rolls down this time.

                                 
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