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Probability of a Rollover in Powerball.

Topic closed. 310 replies. Last post 10 years ago by Prob988.

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dvdiva's avatar - 8ball

United States
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Posted: October 13, 2005, 9:40 pm - IP Logged

I'm curious how a spike in sales would alter the odds. When the prize gets this big sales really start becoming a variable. Sales for the last drawing was 63 mil. It could be 85 - 110 million this time. I always look forward to your projections - keep up the excellent work.


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    Posted: October 14, 2005, 1:06 pm - IP Logged

    This is a great probability you've shown Prob988. I hope these jackpots continue to spike upward. I hope the PB jackpot rolls into Wednesday as the largest jackpot PB has ever had in their history.

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      New Jersey
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      Posted: October 15, 2005, 1:09 pm - IP Logged

      The jackpot has risen again to a cash value of $146M, still well short of Jack's jackpot, but nonetheless impressive.    From the size of the new jackpot, it would seem that the lottery is predicting $76.7M in sales for this drawing.  The probability for various numbers of winners is now:

       

      059.16%
      131.06%
      28.15%
      31.43%
      40.19%
      50.02%


       

      There is still a better than even chance that the jackpot will roll over tonight.  Of course it is very possible that there will be one or more winners, as well.

      If the jackpot rolls, it will be interesting to see if the annuity prize results in the institution of a cap before the cash value of Jack's jackpot is reached.  I'm not sure of the Powerball policy on this matter.

       

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        Morrison, IL
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        Posted: October 15, 2005, 9:18 pm - IP Logged

          If the cap works already it will rise only to $365 million.


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          Posted: October 16, 2005, 2:54 am - IP Logged

          Prob988, the PB jackpot on Saturday 10/15 has rolled over to 340 million for Wednesday 10/19. Do you think that the PB jackpot will roll and shatter the North American jackpot record for Saturday 10/22?

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            Posted: October 16, 2005, 3:40 am - IP Logged
               

            Prob988, the PB jackpot on Saturday 10/15 has rolled over to 340 million for Wednesday 10/19. Do you think that the PB jackpot will roll and shatter the North American jackpot record for Saturday 10/22?

            Well, it's still got a bit to go to break the all-important cash value record (which I don't exactly know where it stands). But when it comes the deceptor's pot, yes just one more roll.

            (insert signature here)

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              New Jersey
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              Posted: October 16, 2005, 8:56 am - IP Logged

              The cash value of the Powerball jackpot that just rolled over was, based on sales, was $151M.  The new cash value is $164.4M.  Although the new cash jackpot is not as high as Jack's jackpot, $170.5M, the Powerball marketing department's annuity value to click in the lottery cap.  This means that it is impossible to match Jack's jackpot on this drawing.  The Powerball will be distributing, it says, about $13,000,000 of cash value to the second tier 5+0 prizes.

              The combined cash value of the 5+0 and cash value suggests that the Powerball group is predicting sales of around $90M on this draw.  The sales for the last drawing were $93M.  This is, of course, subject to change:  The 5+0 jackpot may rise.

              I will use the $93M figure to estimate the probabilities of yet another rollover.  The likelihood of various numbers of winners is given in the following table:

              052.91%
              133.68%
              210.72%
              32.27%
              40.36%
              50.05%


              On this basis, there is a slightly better than even chance of the jackpot rolling over yet again, in which case Jack's jackpot will finally be matched.

               

               

               

                Bradly_60's avatar - disney37
                Atlantic Mine, Michigan
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                Posted: October 16, 2005, 11:33 am - IP Logged

                How do you come up with that number?  I mean they sold over $93,000,000 tickets and even if you take into consideration the fact that some of those were repeats there are only a total of 146 million combos.  That means that 63% of the combos were bought.

                Brad

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                  New Jersey
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                  Posted: October 16, 2005, 2:44 pm - IP Logged

                  How do you come up with that number?  I mean they sold over $93,000,000 tickets and even if you take into consideration the fact that some of those were repeats there are only a total of 146 million combos.  That means that 63% of the combos were bought.

                  Brad

                  These figures are determined by a Poisson distribution which is well known to model random distributions of finite, if large numbers, of spaces.    This distribution is appropriate since there is a finite probability that someone will buy a ticket number that has already been purchased.   As more tickets are purchased, the probability of buying the same set of numbers as one that is already covered becomes increasingly higher.

                  In my figures I report only those probabilities that are greater than 1/100th of a percent.  However this does not imply that even more winners can occur than those which I report.  Undoubtedly from time to time there are some combinations that are covered by ten or more different tickets.  The probability of these probably rare occurances coming up as a winning number is low, but it is NOT impossible, any more than it is impossible to win with just one ticket purchase.

                  With 93 million in sales, the odds of there being 10 winners is roughly 1 in 160 million, which is roughly comparable to the odds of winning the lottery itself.  If it were impossible to beat such odds, if it were impossible to win the lottery, no one would buy tickets.

                  The distribution of probabilities for various numbers of winners are given here for reference.  I have not used percentages here, but decimal fractions, expressed in scientific notation.

                   

                  64.89E-05
                  74.44E-06
                  83.54E-07
                  92.50E-08
                  101.59E-09


                  The model of simply dividing the number of tickets sold by the odds to determine coverage is a poor one.  Note that the odds of there being seven jackpot winners is the same order of magnitude as a particular ticket winning 5 + 0.

                   

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                    New Jersey
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                    Posted: October 17, 2005, 9:48 am - IP Logged

                    Sales of course could go higher.    If they match the sales from the draw in which Jack won is jackpot, $132M, the probability distribution will look like this.

                     

                    040.52%
                    136.60%
                    216.54%
                    34.98%
                    41.12%
                    50.20%
                    60.03%


                    Probably the real result will be somewhere in between, close to an even or just short of an even chance of another rollover.

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                      New Jersey
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                      Posted: October 17, 2005, 5:22 pm - IP Logged

                      Under the circumstances, readers may be curious about the number of 5+0 winners and the probability of that number of winners occuring.

                      Let's imagine that $132,000,000 tickets sell, as with Jack's jackpot drawing on Christmas day some years ago.  We can estimate the number of 5 + 0 winners.

                      The distribution is much wider than with the jackpot, but the first number of winners that has 0.01% probability is 17 winners.  The most probable number of winners is 35 and 36 which each have a 6.54% probability.  The largest number of winners with a 0.01% probability is 61.

                      The bonus is only paid if someone wins the jackpot.  Were that to happen on this jackpot, 5+0 would most probably pay somewhere around $500,000.

                       

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                        New Jersey
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                        Posted: October 20, 2005, 8:42 am - IP Logged

                        The recent powerball sold $166.5M tickets, a record, and far more than I thought it would.  It did not exceed Jack's record cash jackpot.

                        Based on the final sales, the actual outcome (1 winner) was the most probable, barely beating out another rollover.  The probabilities of various numbers of winners looked like this:

                         

                         

                        032.17%
                        136.49%
                        220.69%
                        37.82%
                        42.22%
                        50.50%
                        60.10%
                        70.02%


                         

                        The new reset value will probably sell around $12M worth of tickets (estimate), and the distribution will look like this:

                        092.04%
                        17.63%
                        20.32%
                        30.01%


                         

                        I have assumed that the new annuity structure will not significantly effect the sales of a first draw.

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                          New Jersey
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                          Posted: October 23, 2005, 2:15 am - IP Logged

                          The powerball has rolled over and the new estimated cash jackpot is $9.6M.  The previous cash jackpot was, based on sales, almost $4.7M.  The lottery apparently anticipates selling about 16.5M tickets.  The probabilities of various numbers of winners is roughly given as follows.

                           

                          089.32%
                          110.09%
                          20.57%
                          30.02%


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                            Morrison, IL
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                            Posted: October 23, 2005, 12:10 pm - IP Logged

                              I don't know if they anticipate selling $16.5 million tickets, but the $5 million rollover that occurred was the MINIMUM rollover.

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                              Posted: October 27, 2005, 1:33 pm - IP Logged

                              The powerball has rolled over and the new estimated cash jackpot is $13M.  The previous cash jackpot was, based on sales, almost $8.7M.  The lottery apparently anticipates selling about 14.4M tickets.  The probabilities of various numbers of winners is roughly given as follows.

                              090.61%
                              18.93%
                              20.44%
                              30.01%


                                 
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