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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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September 4, 2005
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Posted: April 22, 2006, 5:11 am - IP Logged

The MM jackpot has rolled.  The cash value is estimated at 11.2M; the annuity at $20M.  Second draws average 15.9M in sales.  If this is how  many tickets are sold, the probability of various numbers of winners is as follows:

091.35%
18.27%
20.37%
30.01%

Although the long term probability does not give accurate figures for the first several draws, it is a fair estimate over the long run.  Here is what it predicts for jackpot evolution.

 $540,637,881.41$303,777,28550.06%0.46%
$471,833,064.44$265,116,76754.27%0.93%
(Average Annuity, M)$411,051,203.24$230,964,24258.28%1.71%
$315 $357,356,784.52$200,794,05762.07%2.94%
$262 $309,923,380.15$174,141,85465.61%4.73%
$247 $268,020,927.27$150,597,41968.92%7.21%
$217 $231,004,491.68$129,798,37371.98%10.46%
$186 $198,304,341.47$111,424,59074.79%14.54%
$158 $169,417,178.05$95,193,27577.37%19.44%
$135 $143,898,389.76$80,854,60579.72%25.12%
$115 $121,355,208.68$68,187,88981.85%31.51%
$96 $101,440,665.43$56,998,17083.79%38.50%
$81 $83,848,248.83$47,113,22485.53%45.95%
$69 $68,307,188.32$38,380,90787.10%53.73%
$57 $54,578,286.41$30,666,81888.52%61.68%
$46 $42,450,237.14$23,852,22889.79%69.68%
$36 $31,736,373.78$17,832,24990.92%77.61%
$27 $22,271,795.79$12,514,22891.94%85.36%
$17 $13,910,830.81$7,816,31292.84%92.84%

    dvdiva's avatar - 8ball

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    Posted: April 22, 2006, 11:51 pm - IP Logged

    I know it must be a lot of work but have you ever done a meta-analysis of Megamillions. For instance how often a 300 million jackpot should occur every year? I doubt the current run will even make 250 but it would be interesting to know if Mega is behind on large pots or where it should be.

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      New Jersey
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      Posted: April 23, 2006, 10:17 am - IP Logged

      I know it must be a lot of work but have you ever done a meta-analysis of Megamillions. For instance how often a 300 million jackpot should occur every year? I doubt the current run will even make 250 but it would be interesting to know if Mega is behind on large pots or where it should be.

      There are several ways of looking at this problem, some more sophisticated than others.  I don't have time for a very sophisticated analysis, although I have collected extensive data and probably could flesh it out.

      Here is a simple approach that doesn't take much time.   It is of course the nature of probability that one can never claim that there is a certainty of a $300M jackpot - or a $500M jackpot.  It is possible for instance that every jackpot in the future will have a winner.  It is not very probable - but it is possible.    It is also possible that every run will produce a $500M jackpot - but not probable. 

      However we can examine how many runs are required to give us a 99% probability that at least one $300M will occur.  If one examines the average run since the matrix change, we see that on average a run lasts for 7.7 draws.  I round that up to 8 to give me an integer, and using the modeling program, I find that the probability of going from an eighth draw to a 16th draw with a $300M+ jackpot is roughly 11%.  Using these numbers one can show that there is (roughly) a 99% probability that a $300M jackpot will occur within 42 average runs.  Since an average run lasts about a month, this means that one would expect a 99% chance of a $300M jackpot every 3 and half years.

      If one is less stringent, and wishes to know only how many runs would be required for a 50% chance of a $300M jackpot, a similar calculation shows that such an event is at this probability every 6 runs, or every six months.

      Using similar arguments, I estimate that there is a 50% probability of a $500M jackpot every 33 months.  It would take more almost 19 years for there to be a 99% probability of a $500M jackpot.   

      I have no idea how long this matrix will last, or how long the game will be meaningful, but if things stay the same, there is a decent chance of a $500M jackpot in the next several years.

       

       

       

       

       

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        Morrison, IL
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        Posted: April 23, 2006, 1:10 pm - IP Logged

        If every Powerball or Mega Millions jackpot in the future had a winner, they'd go out of business because the cash values of their guaranteed jackpots are always larger than 30.28855% (PB) or 31.8% (MM) of their first draw of the run sales. This is especially true of Powerball.

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          Posted: April 23, 2006, 1:53 pm - IP Logged

          If every Powerball or Mega Millions jackpot in the future had a winner, they'd go out of business because the cash values of their guaranteed jackpots are always larger than 30.28855% (PB) or 31.8% (MM) of their first draw of the run sales. This is especially true of Powerball.

          It is possible for this to happen nonetheless.  It is however, extremely improbable.  The odds against 10 first round wins in a row is about 1 in ten billion.  This is more than 50 times more unlikely than a single ticket winning the Mega Millions.

          However extremely improbable events do occur.  Given that human beings have 46 chromosomes, each with a probability of 1/2 of being included in a child, ignoring even the probability of any two people meeting and having a relationship, one can see that the odds against any individual existing at all is more than one in 70 trillion.  Thus the Mega Millions is 700 times more likely to be won ten times in a row than it was for either you or I to be born.

          The odds of 6 billion people all inhabiting the same planet with the same genetic code, the case which now exists, is roughly one chance in a number with over 80 billion zeros after it.

          The governments are making a bet in offering minimal prizes.  They are satisfied that they will win this bet, and so far they have been doing so on a grand scale.  But it is not a certainty that they will always win. 

           


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            Posted: April 24, 2006, 1:40 pm - IP Logged

            I think MM will continue to roll Prob988. At this point, it's very hard to win in both MM and PB. The chances of either game go out of business is less than .1%.

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              Morrison, IL
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              Posted: April 24, 2006, 11:10 pm - IP Logged

              Even when Powerball got its lucky string of winners back in winter of 2005, it was still making profits.

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                Posted: April 25, 2006, 11:45 am - IP Logged

                Even when Powerball got its lucky string of winners back in winter of 2005, it was still making profits.

                The Powerball reset cash value is $6.9 million dollars.  On average (under the new matrix), Powerball sells 14.6 million tickets.  Even deducting for smaller prizes, sales commissions and administration, I'm sure they make a profit in the rare cases where the first round drawing produces a winner.  It may not be a huge profit such as that provided by later drawings in a particular jackpot run, but it is nonetheless, a profit.

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                  Morrison, IL
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                  Posted: April 25, 2006, 12:27 pm - IP Logged

                  They may have money left over, but it's not the required 50% less expenses, so that causes them to just deduct from future jackpots, like they do when they get an excessive amount of set prize winners, such as those two times last year when they got 86+ second place winners. Someone has to pay for "guaranteed" prizes, and I guess the states will be damned if they have to take it out of their program funds, so ultimately the nonguaranteed jackpot winners pay for them.

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                    Posted: April 25, 2006, 7:09 pm - IP Logged

                    They may have money left over, but it's not the required 50% less expenses, so that causes them to just deduct from future jackpots, like they do when they get an excessive amount of set prize winners, such as those two times last year when they got 86+ second place winners. Someone has to pay for "guaranteed" prizes, and I guess the states will be damned if they have to take it out of their program funds, so ultimately the nonguaranteed jackpot winners pay for them.

                    I don't recall those events, but to be frank, I seldom look at the number of lower prize winners.

                    When was there more than 86 second prize winners? 

                     

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                      Posted: April 26, 2006, 9:16 am - IP Logged

                      The MM jackpot has rolled over.  The cash value of the prize is 16.9M and the advertised annuity jackpot is $30M.

                      The average sales for 3rd round drawings is $16.1M.  If this is the number of tickets sold, the probability of various numbers of winners is as follows:

                      091.24%
                      18.36%
                      20.38%
                      30.01%

                      Lottoreport has not yet posted the sales figures for the last drawing.  I will update my program after the figures have been posted, and post the long term probabilities afterwards.


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                        Posted: April 26, 2006, 12:30 pm - IP Logged

                        It will be interesting to see what MM does with sales in next draw. I hope MM continues to roll. If it does, a roll 3 record may occur because of interest rates. However, gas prices may not make a roll 3 record possible in MM.

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                          Posted: April 26, 2006, 5:29 pm - IP Logged

                          It is now possible to post the long term probability function results.  Keep in mind that this function is not usually very accurate in the early draws, but that the deviations tend to cancel out so that the function gives fairly accurate impressions overall.  Overall the function predicts, on average, sales within 1% of the historical data.

                          Here is the long term probability chart:

                           

                           $540,637,881.41$303,777,28550.06%0.50%
                           $471,833,064.44$265,116,76754.27%1.00%
                          (Average Annuity, M)$411,051,203.24$230,964,24258.28%1.84%
                          $315 $357,356,784.52$200,794,05762.07%3.16%
                          $262 $309,923,380.15$174,141,85465.61%5.10%
                          $247 $268,020,927.27$150,597,41968.92%7.77%
                          $217 $231,004,491.68$129,798,37371.98%11.27%
                          $186 $198,304,341.47$111,424,59074.79%15.66%
                          $158 $169,417,178.05$95,193,27577.37%20.93%
                          $135 $143,898,389.76$80,854,60579.72%27.06%
                          $115 $121,355,208.68$68,187,88981.85%33.94%
                          $96 $101,440,665.43$56,998,17083.79%41.47%
                          $81 $83,848,248.83$47,113,22485.53%49.50%
                          $69 $68,307,188.32$38,380,90787.10%57.87%
                          $57 $54,578,286.41$30,666,81888.52%66.43%
                          $46 $42,450,237.14$23,852,22889.79%75.05%
                          $36 $31,736,373.78$17,832,24990.92%83.59%
                          $27 $22,271,795.79$12,514,22891.94%91.94%

                           

                          It would seem about an even bet right now that the annuity jackpot will reach about $100M, roughly $57M cash.

                           

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                            Morrison, IL
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                            Posted: April 26, 2006, 7:13 pm - IP Logged

                            They may have money left over, but it's not the required 50% less expenses, so that causes them to just deduct from future jackpots, like they do when they get an excessive amount of set prize winners, such as those two times last year when they got 86+ second place winners. Someone has to pay for "guaranteed" prizes, and I guess the states will be damned if they have to take it out of their program funds, so ultimately the nonguaranteed jackpot winners pay for them.

                            I don't recall those events, but to be frank, I seldom look at the number of lower prize winners.

                            When was there more than 86 second prize winners? 

                             

                            On June 18, 2005, there were 86 second place winners (but interestingly no jackpot winner), and on March 30, 2005, there were 110 second place winners, thanks to a fortune cookie. The PowerPlay multiplier had also been 5 both times.

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                              New Jersey
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                              Posted: April 26, 2006, 8:42 pm - IP Logged

                              They may have money left over, but it's not the required 50% less expenses, so that causes them to just deduct from future jackpots, like they do when they get an excessive amount of set prize winners, such as those two times last year when they got 86+ second place winners. Someone has to pay for "guaranteed" prizes, and I guess the states will be damned if they have to take it out of their program funds, so ultimately the nonguaranteed jackpot winners pay for them.

                              I don't recall those events, but to be frank, I seldom look at the number of lower prize winners.

                              When was there more than 86 second prize winners? 

                               

                              On June 18, 2005, there were 86 second place winners (but interestingly no jackpot winner), and on March 30, 2005, there were 110 second place winners, thanks to a fortune cookie. The PowerPlay multiplier had also been 5 both times.

                              Wow.

                              That's phenomenal.

                              Thanks. 

                               

                                 
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