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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
United States
Member #21206
September 4, 2005
949 Posts
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Posted: October 7, 2006, 3:52 am - IP Logged

The MegaMillions jackpot has rolled.  The cash value is now $19.8M whereas the advertised annuity value is reported at $34M.

Fourth round drawings in MM sell on average $16.9M in tickets.  If this is how many sell at this time, then the randomized probability distribution for numbers of winners will be as follows:

 

090.82%
18.74%
20.42%
30.01%

 

 

The modelling function calculates jackpots based on sales using the model formula S = C exp(a *N), where S is sales, N is the drawing number and C is a constant.  C and a are parameters determined by least squares approaches using historical average data.  The values of C and a for the most up to date data are respectively $9,807,812.13 and the dimensionless number 0.127.    Jackpots and rollover probabilities are generated from these sales figures, using the most recent cash to annuity ratio.

This function suggests the following evolution probabilities for the jackpot drawing:

 

Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
$598,029,761.25$348,264,39045.07%0.24%
$521,554,735.18$303,728,93449.55%0.54%
$454,174,332.38$264,489,75853.86%1.09%
$394,806,994.92$229,917,01557.98%2.03%
$315 $342,499,786.71$199,455,75861.86%3.50%
$262 $296,413,097.51$172,617,03965.49%5.66%
$247 $255,807,165.86$148,970,05568.88%8.64%
$217 $220,030,204.79$128,135,23772.00%12.54%
$186 $188,507,939.66$109,778,15374.87%17.42%
$158 $160,734,390.16$93,604,14577.49%23.26%
$131 $136,263,748.60$79,353,59579.87%30.02%
$111 $114,703,223.94$66,797,76082.04%37.58%
$94 $95,706,736.97$55,735,10083.99%45.81%
$82 $78,969,365.19$45,988,04285.75%54.54%
$69 $64,222,448.31$37,400,13287.34%63.61%
$56 $51,229,275.90$29,833,51988.75%72.83%
$46 $39,781,287.77$23,166,75090.02%82.06%
$35 $29,694,726.30$17,292,81191.15%91.15%

 

 This function is generally more accurate in the latter drawings, at least until the jackpot gets very high.

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    California
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    October 1, 2006
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    Posted: October 7, 2006, 12:27 pm - IP Logged

    Prob988...thanks for the explanation of how the modeling function calculates jackpots.  Still not sure if I totally understand, but then that's why you are a math genuis!!

    Was wondering, how far back do you go to get the average drawing sales?  Is it a rolling year or is it when California joined and they changed the jackpot minimum level?

    On Wednesday and Saturday mornings after checking the numbers and realizing I don't need to choose between lump sum and annuity, my next step is to look for your post to see what the probablity is for  a winner in the next draw.  Keep up the good work!!


      United States
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      June 2, 2005
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      Posted: October 7, 2006, 1:51 pm - IP Logged

      I think MM jackpots are higher than expected in this current run because of lower gas prices and the interest rates are steady. Also, I think the string of jackpot winners made it possible.

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        New Jersey
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        September 4, 2005
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        Posted: October 7, 2006, 10:14 pm - IP Logged

        Prob988...thanks for the explanation of how the modeling function calculates jackpots.  Still not sure if I totally understand, but then that's why you are a math genuis!!

        Was wondering, how far back do you go to get the average drawing sales?  Is it a rolling year or is it when California joined and they changed the jackpot minimum level?

        On Wednesday and Saturday mornings after checking the numbers and realizing I don't need to choose between lump sum and annuity, my next step is to look for your post to see what the probablity is for  a winner in the next draw.  Keep up the good work!!

        The averages refer to all drawings since the last change of matrices, which coincided with the addition of California.  It would not make sense to refer to a time other than the conditions under which the lottery now operates.

        I'm reasonably competent in mathematics, but I assure you I'm no genius.    This stuff is pretty straight forward.

          dvdiva's avatar - 8ball

          United States
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          September 17, 2003
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          Posted: October 11, 2006, 1:09 am - IP Logged

          The averages refer to all drawings since the last change of matrices, which coincided with the addition of California.  It would not make sense to refer to a time other than the conditions under which the lottery now operates.

          I'm reasonably competent in mathematics, but I assure you I'm no genius.    This stuff is pretty straight forward.

          I would say you are far more than just competent, I always look forward to your pobabilty outcomes.

          Of course when it comes to Mega breaking 100 million vs a bunch of low level winners it always seems to go in one direction. On this run it looks like if it can survive until Nov 17th it would be over 300.

          I wonder if a string of low level wins is more probable than a large jackpot (something I can't answer myself). 

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            New Jersey
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            Posted: October 11, 2006, 2:45 am - IP Logged

            The MM jackpot has rolled over, and the cash value is now $26.2M.  The advertised annuity value is $45M.  The average 5th draw produces 18.6M in sales.  If the next drawing sells an average number of tickets, the randomized Poisson probability for various numbers of winners is as follows:

             

            089.94%
            19.54%
            20.51%
            30.02%

             

            The exponential modelling function gives the following long term probability for the 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
            $598,197,985.38$348,284,16045.06%0.27%
            $521,683,311.40$303,735,61749.54%0.59%
            $454,270,718.29$264,486,50753.86%1.20%
            $394,877,434.97$229,906,41857.97%2.23%
            $315 $342,549,495.33$199,439,92861.85%3.84%
            $262 $296,446,415.76$172,597,69165.49%6.21%
            $247 $255,827,695.43$148,948,56968.87%9.48%
            $217 $220,040,922.46$128,112,71572.00%13.76%
            $186 $188,511,295.01$109,755,46574.87%19.11%
            $158 $160,732,388.89$93,581,96977.49%25.53%
            $131 $136,258,023.41$79,332,44979.88%32.94%
            $111 $114,695,094.94$66,778,03382.04%41.24%
            $94 $95,697,262.93$55,717,07384.00%50.27%
            $82 $78,959,387.01$45,971,91085.76%59.85%
            $69 $64,212,625.89$37,386,01887.34%69.79%
            $56 $51,220,119.29$29,821,49288.76%79.90%
            $46 $39,773,183.49$23,156,83190.02%90.02%

             

            The modeling function is exponential, but the experimental sales, while approximating an exponential function, deviates in a regular way that can be seen graphically.  For drawings 1-4, the exponential function under estimates sales.  For drawing 5, the functional value is close to the experimental value.  Then from drawing 6-13, the function over estimates sales, before beginning to under estimate sales again for drawings 14-17.  For the long term these effects cancel out, giving a useful approximation of how likely large jackpots are.    As it stands now, we would seem to have a roughly even chance of seeing an annuity jackpot greater than $100 million.  Such an event will take another 2 and one half weeks, creating a perception that the jackpot is "on a roll."

            The "average" number of rolls is 7.47.

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              New Jersey
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              Posted: October 11, 2006, 10:26 am - IP Logged

              I would say you are far more than just competent, I always look forward to your pobabilty outcomes.

              Of course when it comes to Mega breaking 100 million vs a bunch of low level winners it always seems to go in one direction. On this run it looks like if it can survive until Nov 17th it would be over 300.

              I wonder if a string of low level wins is more probable than a large jackpot (something I can't answer myself). 

              It depends on what you mean by a "long string of low level wins" and a "large" jackpot.   There are several ways to interpret your question, and I will give you just one based on recent events.

              Between April 21 and August 1 of this year, we had a string of runs in which the jackpot never passed $100M in advertised annuity value.  The lengths of the runs in this period were 8 (about average) followed by below average length runs of 5, 4, 3, 6 and 4 draws.

              Working from my modeling program, I calculate that the probability of having a set exactly consisting of runs with this many drawings (in any order) is 0.11%.    This is smaller than the probability of a jackpot rolling beyond $600M annuity from a given first draw.  Thus the events of this summer were improbable.

              However, we should recognize that there are many other sequences that might constitute a "string of low level wins."  I don't have the time to examine a subset of them, never mind all of the different possible ways in which such a string might occur.

              It is easier to ask the question, how long is it likely to take to have a jackpot that is more than $500M in advertised annuity?    Keep in mind that such a thing can never become a certainty, but if one looks at a longer and longer stretch of time, the probability of such a jackpot becomes more and more likely.   

              Here is a table that indicates a percentage probability of a $500M jackpot and the length of time indicated in draws, weeks, and years that such a jackpot will have the probability indicated.  To convert from draws to weeks, I have used the average value of 7.47 draws/run. 

               (I'm sorry the header ran together, but it's an unavoidable consequence of the cut and paste from excel.)

               

              Odds of $500M jackpot occurring in a periodNumber of runs required to produce this probability of a $500M jackpotNumber of draws required to reach these oddsNumber of weeks required to reach these oddsNumber of years required to reach these odds
              5.00%643220.42
              10.00%1289450.86
              15.00%18137691.32
              20.00%25189941.81
              25.00%332431222.34
              30.00%403021512.90
              35.00%493641823.50
              40.00%584322164.15
              45.00%685062534.86
              50.00%785862935.64
              55.00%906753386.49
              60.00%1047753877.45
              65.00%1198884448.54
              70.00%13610185099.79
              75.00%157117258611.27
              80.00%182136168013.09
              85.00%215160480215.42
              90.00%261194797318.72
              95.00%3392533126724.36

               

               

              This event could occur on the current run, or not at all, but it likely to happen within the next six years.  There's a five percent chance we'll see such a jackpot this year.  But t have a 95% chance, we would need to wait almost 25 years.

              All these events are random.  There is no way to accurately predict exactly when it will happen, but we can get an idea of how likely it is in a given period. 

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                New Jersey
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                Posted: October 14, 2006, 1:59 am - IP Logged

                The MM jackpot has rolled.  The cash value is $31.8M.  The advertised annuity is $55M.

                Average sixth round draws sell 19.5M tickets.  If this is the number of tickets that sell that the probability of various numbers of winners is given in the following table:

                 

                089.49%
                19.94%
                20.55%
                30.02%

                 

                The exponential modeling function gives giving the probability of a number of future rollovers:

                 

                Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                $604,265,875.82$348,276,87845.06%0.30%
                $526,985,059.78$303,735,02549.54%0.66%
                $458,895,952.80$264,490,94053.86%1.33%
                $398,905,301.49$229,914,51057.97%2.47%
                $315 $346,049,882.98$199,450,56961.86%4.26%
                $262 $299,481,039.26$172,609,98165.49%6.89%
                $247 $258,451,050.92$148,961,78868.87%10.53%
                $217 $222,301,131.59$128,126,28972.00%15.28%
                $186 $190,450,850.36$109,768,94574.87%21.23%
                $158 $162,388,812.25$93,595,00677.49%28.35%
                $131 $137,664,447.09$79,344,78179.88%36.59%
                $111 $115,880,775.16$66,789,46582.04%45.80%
                $94 $96,688,033.09$55,727,46683.99%55.83%
                $82 $79,778,058.09$45,981,17285.75%66.47%
                $69 $64,879,339.94$37,394,09287.34%77.51%
                $56 $51,752,661.60$29,828,35288.75%88.75%
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                  New Jersey
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                  Posted: October 19, 2006, 7:07 am - IP Logged

                  The MM jackpot has been won and the jackpot has been reset to $6.9M cash and $12M as an advertised annuity.

                  The average number of sales for a first round drawing is $14.3M.  If this is how many tickets sell this draw, the probability distribution for numbers of winners is as follows:

                   

                  091.67%
                  17.97%
                  20.35%
                  30.01%

                   

                  The exponential modeling function gives giving the probability of a number of future rollovers:

                   

                  Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                  $605,709,951.96$348,283,22245.05%0.20%
                  $528,217,873.10$303,725,27749.53%0.44%
                  $459,946,650.39$264,469,32453.85%0.88%
                  $399,799,085.40$229,884,47457.97%1.63%
                  $315 $346,808,536.37$199,414,90861.85%2.82%
                  $262 $300,123,383.17$172,570,94565.49%4.56%
                  $247 $258,993,340.74$148,921,17168.87%6.96%
                  $217 $222,757,401.18$128,085,50672.00%10.10%
                  $180 $190,833,210.52$109,729,09674.87%14.03%
                  $153 $162,707,709.68$93,556,93377.49%18.74%
                  $131 $137,928,888.93$79,309,11179.88%24.19%
                  $111 $116,098,523.62$66,756,65182.04%30.28%
                  $94 $96,865,774.24$55,697,82084.00%36.91%
                  $82 $79,921,548.00$45,954,89085.76%43.94%
                  $69 $64,993,531.36$37,371,28187.34%51.23%
                  $52 $51,841,813.61$29,809,04388.76%58.66%
                  $43 $40,255,031.27$23,146,64390.03%66.08%
                  $33 $30,046,971.18$17,277,00891.16%73.40%
                  $25 $21,053,577.86$12,105,80792.17%80.52%
                  $16 $13,130,316.97$7,549,93293.07%87.36%
                  $12 $6,149,852.43$3,536,16593.87%93.87%
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                    Morrison, IL
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                    Posted: October 19, 2006, 7:58 am - IP Logged

                    One more jackpot winner before 2007, and Mega Millions will tie Powerball's record of fourteen jackpots awarded in one year from either multi-state game. Powerball set that record just last year (2005).

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                      New Jersey
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                      Posted: October 19, 2006, 2:30 pm - IP Logged

                      One more jackpot winner before 2007, and Mega Millions will tie Powerball's record of fourteen jackpots awarded in one year from either multi-state game. Powerball set that record just last year (2005).

                      That is a probable outcome.

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                        New Jersey
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                        Posted: October 21, 2006, 8:53 am - IP Logged

                        The MM jackpot has rolled.  The cash value is $9.2M; the advertised annuity is $16M.  Second round jackpots sell on average, 15.0M in tickets.

                        If an average amount is sold, the randomized probability of various numbers of winners will be as follows:

                         

                        091.81%
                        17.84%
                        20.34%
                        30.01%

                         

                        The long term 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
                        $606,150,102.40$348,536,30944.94%0.21%
                        $528,420,917.98$303,842,02849.44%0.46%
                        $459,968,197.32$264,481,71353.78%0.94%
                        $399,684,855.87$229,818,79257.91%1.74%
                        $315 $346,595,932.29$199,292,66161.81%3.01%
                        $262 $299,842,820.40$172,409,62265.46%4.87%
                        $247 $258,669,383.03$148,734,89568.86%7.44%
                        $217 $222,409,722.92$127,885,59171.99%10.80%
                        $180 $190,477,413.30$109,524,51374.87%15.00%
                        $153 $162,356,013.51$93,354,70877.50%20.03%
                        $131 $137,590,716.68$79,114,66279.90%25.85%
                        $111 $115,780,994.14$66,574,07282.07%32.35%
                        $94 $96,574,117.66$55,530,11884.02%39.42%
                        $82 $79,659,454.78$45,804,18685.79%46.92%
                        $69 $64,763,444.97$37,238,98187.37%54.69%
                        $52 $51,645,175.36$29,695,97688.79%62.59%
                        $43 $40,092,484.42$23,053,17990.06%70.50%
                        $33 $29,918,530.68$17,203,15591.19%78.28%
                        $25 $20,958,771.00$12,051,29392.20%85.84%
                        $16 $13,068,299.34$7,514,27293.10%93.10%
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                          New Jersey
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                          Posted: October 25, 2006, 2:04 am - IP Logged

                          The MM jackpot has rolled.  The cash value is $13.8M; the advertised annuity is $24M.  Third round jackpots sell on average, 15.9M in tickets.

                          If an average amount is sold, the randomized probability of various numbers of winners will be as follows:

                           

                          091.31%
                          18.30%
                          20.38%
                          30.01%

                           

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

                           

                           

                          Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                          $603,752,809.51$348,586,94844.92%0.22%
                          $526,298,546.51$303,867,41349.42%0.50%
                          $458,092,860.12$264,487,70053.76%1.01%
                          $398,031,406.12$229,810,19957.90%1.87%
                          $315 $345,141,706.63$199,273,43261.80%3.23%
                          $262 $298,567,404.37$172,382,96765.46%5.23%
                          $247 $257,554,397.07$148,703,41068.85%7.99%
                          $217 $221,438,627.53$127,851,35671.99%11.60%
                          $180 $189,635,331.55$109,489,18274.87%16.12%
                          $153 $161,629,569.87$93,319,58077.50%21.53%
                          $131 $136,967,890.52$79,080,74079.90%27.78%
                          $111 $115,250,986.87$66,542,11782.07%34.77%
                          $94 $96,127,232.26$55,500,69284.03%42.36%
                          $82 $79,286,986.68$45,777,69085.79%50.41%
                          $69 $64,457,583.32$37,215,68287.38%58.76%
                          $52 $51,398,913.66$29,676,03788.80%67.25%
                          $43 $39,899,539.79$23,036,67890.07%75.74%
                          $33 $29,773,270.98$17,190,10491.20%84.09%
                          $25 $20,856,148.96$12,041,65292.21%92.21%
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                            New Jersey
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                            Posted: October 28, 2006, 10:16 am - IP Logged

                            The MM jackpot has rolled.  The cash value is $18.7M; the advertised annuity is $32M.  Third round jackpots sell on average, 16.8M in tickets.

                            If an average amount is sold, the randomized probability of various numbers of winners will be as follows:

                             

                             

                            090.85%
                            18.71%
                            20.42%
                            30.01%

                             

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

                             

                             

                            Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                            $596,529,839.22$348,597,12544.91%0.24%
                            $519,994,776.50$303,871,94849.42%0.54%
                            $452,599,655.73$264,487,92453.76%1.09%
                            $393,252,970.14$229,807,20457.89%2.03%
                            $315 $340,993,562.49$199,268,11361.80%3.51%
                            $262 $294,975,058.48$172,376,05065.46%5.67%
                            $247 $254,452,159.23$148,695,48168.85%8.67%
                            $217 $218,768,570.67$127,842,88371.99%12.59%
                            $180 $187,346,374.42$109,480,53874.87%17.48%
                            $153 $159,676,668.06$93,311,05377.51%23.35%
                            $131 $135,311,323.14$79,072,55479.90%30.13%
                            $111 $113,855,727.43$66,534,44182.07%37.71%
                            $94 $94,962,393.93$55,493,64984.03%45.95%
                            $82 $78,325,333.12$45,771,36785.79%54.68%
                            $69 $63,675,097.23$37,210,13587.38%63.73%
                            $52 $50,774,416.39$29,671,30088.80%72.94%
                            $43 $39,414,355.88$23,032,76490.07%82.14%
                            $33 $29,410,932.32$17,187,01491.20%91.20%
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                              New Jersey
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                              Posted: November 1, 2006, 4:52 am - IP Logged

                              The MM jackpot has rolled.  The cash value is $24.6M ; the advertised annuity is $42M.  Third round jackpots sell on average, 18.7M in tickets.

                              If an average amount is sold, the randomized probability of various numbers of winners will be as follows:

                               

                              089.92%
                              19.55%
                              20.51%
                              30.02%

                               

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

                               

                              Historical Average Advertised Annuity JackpotsModel Calculated Annuity PrizeModel Calculated Cash ValueSingle Draw Calculated Model Rollover ProbabilityOverall Calculated Model Rollover Probability
                              $596,582,280.77$348,627,77044.90%0.27%
                              $520,012,796.63$303,882,47849.41%0.59%
                              $452,591,561.86$264,483,19453.75%1.20%
                              $393,225,576.19$229,791,19657.89%2.23%
                              $315 $340,952,427.13$199,244,07561.79%3.84%
                              $262 $294,924,687.83$172,346,61465.45%6.22%
                              $247 $254,396,178.95$148,662,76768.85%9.51%
                              $217 $218,709,872.02$127,808,58171.99%13.81%
                              $180 $187,287,238.00$109,445,98074.87%19.18%
                              $153 $159,618,868.42$93,277,27677.51%25.62%
                              $131 $135,256,217.13$79,040,35279.90%33.05%
                              $111 $113,804,328.68$66,504,40582.07%41.36%
                              $94 $94,915,435.50$55,466,20884.03%50.39%
                              $82 $78,283,320.02$45,746,81585.80%59.97%
                              $69 $63,638,350.49$37,188,66187.38%69.90%
                              $52 $50,743,109.78$29,653,00588.80%79.99%
                              $43 $39,388,546.52$23,017,68290.07%90.07%
                                 
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