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Probability of a MegaMillions rollover.

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

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dvdiva's avatar - 8ball

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Posted: February 28, 2007, 9:10 am - IP Logged

I'm curious at what sales point does the probability of a rollover hit 50%.

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    NY
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    Posted: February 28, 2007, 3:41 pm - IP Logged

    I'm curious at what sales point does the probability of a rollover hit 50%.

    It happens when exactly 87,855,768 of the possible combinations have been played, but what that is in terms of actual sales is difficult to say.  It doesn't even come close to happening with sales of "only" 100 to 110 million tickets as can be expected for the current drawing. Selling 50% of the possible combinations would require selling about 160 million tickets if the combinations were all picked randomly. Since a lot of people don't pick their combinations randomly there are even more repeats than would result from simple probability. Here's a simplified explanation.

    Let's imagine that the first 175,712 tickets had no repeated combinations. That would cover 1% of the possible combinations. With 1% of the combinations already played we would expect that 1 of the next 100 tickets would be a repeated combination, so selling 100 more tickets only increases the number of combinations that have been played by 99. Selling another 175,712 tickets would add about 173,955 new combinations and 1757 would be repeats. After 351,423 combinations have been played 2% of the combinations will have been played so only 98 of the next 100 tickets will be new combinations. After527,136 (3%) combinations have been played 3% of subsequent tickets would be repeats, and so on.

     To simplify things we can figure that for the first 1% of combinations every ticket would be a new combination and we'd have 100% efficiency. For the next 1% we'd only have 99% efficiency, 98% for the  3rd 1%, and so on. 100% is 1.00, 99% is .99, 98% is .98, and so on. That means that for successive batches of  175,712 tickets sold we'd get  (175,712 X 1.00) + (175,712 X .99) + (175,712 X .98) + (175,712 x .97) etc. new combinations. That's the same as 175,712 X (1.00 + .99 + .98 + .97 ...)

    If 87,855,768 tickets are sold we'd  get 175,712 x (1.00 + .99 + .98 + .97 ... + .53 + .52 + .51) which comes to  175,712 X  37.75. That means we'd only expect 37.75% of the combinations to have been played instead of 50%.  That's a simplification and 37.75% isn't quite right, but nobody wants me to start using calculus.

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      New Jersey
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      Posted: February 28, 2007, 10:18 pm - IP Logged

      The field is 50% covered at about 122 million tickets sold, according to the Poisson distribution.

      The MM jackpot, has of course, rolled over.  The cash value is $158M.  The advertised annuity is $267M.

      The average number of tickets sold on 15th drawings is $72.0M.  However there are only three such drawings involved in this average, but the last of them produced sales of $96M.  This series of drawings from early in 2006, ending in February, closely followed the tragectory of the current run, so I am going to estimate that ticket sales will be 96M for the purpose of estimating the probability of various numbers of winners.

      If 96M tickets are sold, the probability of various numbers of winners will be as follows:

       

      057.91%
      131.64%
      28.64%
      31.57%
      40.21%
      50.02%

       

      The long term exponential modeling function suggests the following:

       

       

      Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
      $596,391,147.72$354,361,32443.90%1.75%
      $518,964,637.44$308,356,34848.50%4.00%
      $450,923,597.65$267,927,99352.95%8.24%
      $391,130,342.35$232,400,27457.19%15.56%
      $315 $338,585,093.34$201,179,14761.20%27.21%
      $262 $292,409,263.31$173,742,57564.95%44.45%
      $247 $251,830,765.36$149,631,80468.44%68.44%

       

      The function predicts sales of around 75M, and I think the modeling function is going to lag from the actual values.  However it crudely suggests a roughly 40% chance of two more rollovers.

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        New Jersey
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        Posted: March 3, 2007, 8:58 am - IP Logged

        The $202M cash, $340M jackpot changes the rules, and my estimates will be crude and somewhat ad hoc.  For what it's worth, here is what I see for the current drawing in terms of winner distribution:

         

        046.54%
        135.60%
        213.61%
        33.47%
        40.66%
        50.10%
        60.01%

         

         

        Long term - and probably it won't go much further:

         

        Model Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
        $699,205,759.87$416,938,49231.96%1.37%
        $592,314,865.81$353,199,13136.85%4.29%
        $498,758,904.27$297,411,43141.73%11.63%
        $416,874,305.83$248,583,39946.54%27.87%
        $345,205,035.88$205,846,79959.88%59.88%
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          New Jersey
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          Posted: March 3, 2007, 1:28 pm - IP Logged

          In my previous analysis, I used a figure of $134M for sales - based on some extrapolation of sales figures from the previous draws in this run.

          However the lottery is apparently estimating - until the jackpot is announced as being higher - sales of about 116M.    The lottery's figure will give the following rollover probability:

           

           

          051.45%
          134.19%
          211.36%
          32.52%
          40.42%
          50.06%
          60.01%

           

           

          I think in any case, there is a fairly decent chance - close to even odds - of this thing rolling again, but as is always the case, anything could happen.

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            Posted: March 5, 2007, 6:15 pm - IP Logged

            With the jackpot rising, it doesn't seem that my 134M sales figure was that far off.

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              New Jersey
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              Posted: March 6, 2007, 8:12 am - IP Logged

              Actually, if the jackpot stays at $212M cash/$355M annuity, 148M tickets will be sold.  This would lead to the following distribution of probabilty for various numbers of winners:

               

              043.00%
              136.29%
              215.31%
              34.31%
              40.91%
              50.15%
              60.02%

               

              It seems thus that while a rollover is the single most probable outcome, one or more winners combined is more likely.    Thus you'd be better guessing that there will be at least one winner.  Note that there is a 1% chance of 4 or more winners.

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                Posted: March 6, 2007, 11:45 am - IP Logged

                Prob988....thanks for the updates!!!!!

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                  Posted: March 6, 2007, 12:29 pm - IP Logged

                  We're closing in on $180 milion in sales. I'm going to have to  forget probability and just cross my fingers that it rolls.  This roll has seen more than a 60% jump in sales over the previous  jackpot, and a similar increase for another roll would mean a single winner would also be a matter of luck rather than probability.

                  FWIW, a similar jump in sales would push the annuity value to about 525  and the cash to about 310.

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                    New Jersey
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                    Posted: March 6, 2007, 1:12 pm - IP Logged

                    $180M in sales would be close to the odds against winning the lottery.  In this case the rollover probability falls to 33%.    This would still not eliminate the probability of a rollover, but of course it would make the likelihood of a winner 67%.

                    A rollover is still possible.  Unlikely, but possible.  I'd like to see it, just to see everyone go crazy.  Of course I wouldn't like to see it as much as I would like to win the existing jackpot. 

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                      Posted: March 6, 2007, 5:59 pm - IP Logged

                      Probably the last update.  A $221M cash prize suggests sales of 177M.    Probability for various numbers of winners:

                       

                      036.50%
                      136.79%
                      218.54%
                      36.23%
                      41.57%
                      50.32%
                      60.05%
                      70.01%

                       

                       

                      The most probable outcome is now one winner.  There still, however, is a decent chance of a rollover.    A rollover is more unlikely than ever, but it is possible.

                        dvdiva's avatar - 8ball

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                        Posted: March 6, 2007, 10:01 pm - IP Logged

                        If it rolls at this point I will be shocked. I'm still shocked that it would have such a high chance of a rollover. If it does roll I'm curious if two winners would be more likely than one winner.

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                          Posted: March 6, 2007, 10:23 pm - IP Logged

                          One thing that determines the # of winners is the kind of numbers that come out, if they were system-friendly or if they are low numbers (ages of children, birthdays, etc.) that increases the chance of multiple winners. A while back 100 people came one # away from sharing the PB jackpot 100-ways because they used the same fortune cookie numbers that came out all but the bonus ball.

                          Always when they did the drawing on Times Square someone would win somewhere...

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                            Posted: March 8, 2007, 7:52 am - IP Logged

                            The final sales for the recent drawing were 212M:

                             

                            This gives this kind of probability:

                             

                            029.78%
                            136.07%
                            221.85%
                            38.82%
                            42.67%
                            50.65%
                            60.13%
                            70.02%

                             

                            There was about a 1 in 5 chance of the result.

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                              Posted: March 11, 2007, 3:44 pm - IP Logged

                              The MM jackpot has rolled over.    The cash value is $11.3M, the advertised annuity, $19M.  Second round drawings produce sales typically of 14.8M in sales.  If this is how many tickets sell, the probability of various numbers of winners will be as follows:

                               

                              082.90%
                              115.55%
                              21.46%
                              30.09%

                               

                              The long term model, which ignores minimums, predicts the following 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
                              $740,676,398.87$440,623,55630.84%0.04%
                              $630,192,280.45$374,897,27536.51%0.13%
                              $535,549,706.69$318,595,02542.18%0.35%
                              $454,477,269.35$270,365,56147.74%0.84%
                              $315 $385,029,241.51$229,051,38353.08%1.75%
                              $262 $325,538,880.43$193,660,95558.13%3.30%
                              $247 $274,578,425.96$163,344,91462.83%5.67%
                              $217 $230,924,834.48$137,375,67767.16%9.02%
                              $180 $193,530,426.02$115,129,98771.10%13.44%
                              $153 $161,497,740.14$96,073,95274.67%18.90%
                              $130 $134,057,996.94$79,750,22777.86%25.31%
                              $109 $110,552,646.52$65,767,04880.70%32.51%
                              $92 $90,417,563.82$53,788,81983.22%40.28%
                              $81 $73,169,509.74$43,528,06485.44%48.40%
                              $67 $58,394,533.45$34,738,52787.39%56.64%
                              $55 $45,738,037.62$27,209,26089.10%64.81%
                              $45 $34,896,268.17$20,759,56290.59%72.74%
                              $34 $25,609,024.17$15,234,64191.88%80.30%
                              $25 $17,653,413.11$10,501,90093.00%87.40%
                              $16 $10,838,501.45$6,447,75493.97%93.97%
                                 
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