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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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 Posted: November 4, 2006, 8:03 am - IP Logged

The MM jackpot has rolled.  The cash value is \$30.4M ; the advertised annuity is \$52M.  Sixth round jackpots sell on average, 19.3M in tickets.

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

 0 89.56% 1 9.88% 2 0.54% 3 0.02%

The long term probability model function is within 3% of actual events.  It suggests that the probability of the jackpot growing is as follows:

 Historical Average Advertised Annuity Jackpots Model Calculated Annuity Prize Model Calculated Cash Value Single Draw Calculated Model Rollover Probability Overall Calculated Model Rollover Probability \$594,392,245.97 \$348,633,914 44.89% 0.29% \$518,095,732.34 \$303,883,074 49.40% 0.66% \$450,916,084.13 \$264,479,626 53.74% 1.33% \$391,763,903.80 \$229,784,597 57.88% 2.47% \$315 \$339,679,968.69 \$199,235,366 61.79% 4.27% \$262 \$293,819,676.11 \$172,336,541 65.45% 6.91% \$247 \$253,439,347.10 \$148,651,925 68.85% 10.56% \$217 \$217,884,166.82 \$127,797,444 71.99% 15.33% \$180 \$186,577,565.91 \$109,434,918 74.87% 21.30% \$153 \$159,011,870.77 \$93,266,578 77.51% 28.44% \$131 \$134,740,070.99 \$79,030,234 79.90% 36.70% \$111 \$113,368,570.60 \$66,495,027 82.08% 45.92% \$94 \$94,550,805.43 \$55,457,684 84.04% 55.95% \$82 \$77,981,623.11 \$45,739,221 85.80% 66.58% \$69 \$63,392,334.73 \$37,182,042 87.39% 77.60% \$52 \$50,546,357.72 \$29,647,383 88.80% 88.80%
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 Posted: November 4, 2006, 11:38 am - IP Logged

Prob988...quick question if I may.  How did you calculate the figure for 0 winners, 89.56% in this instance?

I tried taking the average number of tickets sold (19.3m) divided by total odds (175,711,536) and came up with a different number.  What factors should I be using?

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 Posted: November 4, 2006, 1:41 pm - IP Logged

I use what is called a Poisson distribution, assuming that all purchases are effectively random.

When the number of tickets purchased is very small compared to the odds, a Poisson distribution will give numbers that are close to the numbers obtained by dividing the number of tickets purchased by the probability.  However as the number of tickets purchased increases, there is a greater probability that each new purchaser will purchase a ticket with a set of numbers that has been already purchased by someone else. We all are intuitively aware that the larger the number of tickets that are sold, the more frequently we will see jackpots won by more than one person.  These possibilities are measured by Poisson calculations.

The mathematics of Poisson distributions are available on the internet and in many other places and I encourage you to investigate them.

I have built a series of spreadsheets that do all of my calculations, and produce the results that I provide here from them.    I have arranged it so that I merely enter the sales for each drawing in a box on these spreadsheets and all of the values pop out.  I then cut and paste them here for informational purposes.

Note that I only report values that have a probability of 1 hundredth of one percent.    The probability of there being six winners in the upcoming draw is not zero.    It is possible that there will be six winners, not very likely, but possible.

Sources of error in my calculation include the fact that purchases are not always random.  For instance people play their birthday numbers frequently, which tends to skew the probability of rollovers in someway by creating more doubly covered sets of numbers (those with all of the numbers being less than 31) and creating more uncovered sets  (those which do not represent birthdays.)  There are undoubtedly other such effects, however the Poisson calculations are a useful first approximation.

The long term probabilities I post are based on a mathematical technique known as curve fitting.  I have graphed the sales figures for various jackpot sizes, made an estimate of the type of curve that fits, for MM an exponential function, and for PB a linear function that switches at a certain point to an exponential function, and have calculated certain constants.  These constants are automatically updated for each drawing by a series of linked spreadsheets.

I hope you find them useful and I hope you find these comments useful.

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 Posted: November 8, 2006, 4:00 am - IP Logged

The MM jackpot has rolled.  The cash value is \$37.1M ; the advertised annuity is \$63M.  Seventh round jackpots sell on average, 21.7M in tickets.

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

 0 88.39% 1 10.90% 2 0.67% 3 0.03%

The long term probability modelling function is pretty much on the money right now.  It predicts as follows:

 Historical Average Advertised Annuity Jackpots Model Calculated Annuity Prize Model Calculated Cash Value Single Draw Calculated Model Rollover Probability Overall Calculated Model Rollover Probability \$592,026,467.62 \$348,637,809 44.89% 0.33% \$516,018,875.83 \$303,877,782 49.40% 0.74% \$449,095,861.20 \$264,467,563 53.74% 1.50% \$390,171,617.16 \$229,767,730 57.88% 2.78% \$315 \$338,290,114.89 \$199,215,290 61.79% 4.81% \$262 \$292,609,592.02 \$172,314,538 65.45% 7.78% \$247 \$252,388,895.26 \$148,629,016 68.85% 11.89% \$217 \$216,975,455.44 \$127,774,435 71.99% 17.27% \$180 \$185,794,699.74 \$109,412,434 74.88% 23.99% \$153 \$158,340,729.41 \$93,245,096 77.51% 32.03% \$131 \$134,168,111.72 \$79,010,110 79.91% 41.33% \$111 \$112,884,652.93 \$66,476,518 82.08% 51.72% \$94 \$94,145,035.08 \$55,440,965 84.04% 63.01% \$82 \$77,645,213.28 \$45,724,403 85.80% 74.98% \$69 \$63,117,482.71 \$37,169,184 87.39% 87.39%
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 Posted: November 11, 2006, 1:24 am - IP Logged

The MM jackpot has rolled.  The cash value is \$44.3M ; the advertised annuity is \$75M.  Eighth round jackpots sell on average, \$23.2M in tickets.    If the current drawing results in average sales the probability of various numbers of winner is as follows:

 0 87.61% 1 11.59% 2 0.77% 3 0.03%

The long term model, a curve fit exponential function now overestimates the jackpot by 3%.  It is nonetheless useful for estimating the long term probability of the lottery going to higher jackpots.

Here are the values:

 Historical Average Advertised Annuity Jackpots Model Calculated Annuity Prize Model Calculated Cash Value Single Draw Calculated Model Rollover Probability Overall Calculated Model Rollover Probability \$590,231,123.34 \$348,629,850 44.88% 0.38% \$514,437,385.11 \$303,861,015 49.39% 0.85% \$447,705,186.67 \$264,444,530 53.74% 1.71% \$388,951,173.30 \$229,740,493 57.88% 3.19% \$315 \$337,221,511.06 \$199,185,506 61.79% 5.50% \$262 \$291,676,401.83 \$172,283,528 65.45% 8.91% \$247 \$251,576,449.81 \$148,597,823 68.85% 13.61% \$217 \$216,270,657.82 \$127,743,869 71.99% 19.76% \$180 \$185,185,858.84 \$109,383,114 74.88% 27.45% \$153 \$157,817,410.96 \$93,217,484 77.51% 36.66% \$131 \$133,721,004.88 \$78,984,540 79.91% 47.30% \$111 \$112,505,450.80 \$66,453,220 82.08% 59.19% \$94 \$93,826,327.73 \$55,420,084 84.04% 72.11% \$82 \$77,380,392.00 \$45,706,018 85.81% 85.81%
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 Posted: November 15, 2006, 11:38 pm - IP Logged

The MM jackpot has been won.    The cash prize is reset to \$7.1M and the advertised annuity is \$12M.  Average 1st draws produce 15.2M in sales.    Selling an average number of tickets would result in the following distribution for the probability of various numbers of winners:

 0 91.71% 1 7.93% 2 0.34% 3 0.01%

Here is the long term probability, which ignores minimums:

 Historical Average Advertised Annuity Jackpots Model Calculated Annuity Prize Model Calculated Cash Value Single Draw Calculated Model Rollover Probability Overall Calculated Model Rollover Probability \$589,189,237.47 \$348,603,632 44.87% 0.20% \$513,513,969.88 \$303,829,099 49.39% 0.44% \$446,888,401.17 \$264,408,971 53.73% 0.88% \$388,230,314.51 \$229,702,936 57.88% 1.64% \$315 \$336,586,910.96 \$199,147,256 61.79% 2.83% \$262 \$291,119,332.98 \$172,245,605 65.45% 4.58% \$247 \$251,089,038.57 \$148,561,014 68.85% 7.00% \$217 \$215,845,805.02 \$127,708,768 72.00% 10.17% \$180 \$184,817,167.09 \$109,350,157 74.88% 14.13% \$153 \$157,499,118.36 \$93,186,978 77.52% 18.87% \$131 \$133,447,924.48 \$78,956,689 79.91% 24.34% \$111 \$112,272,915.51 \$66,428,142 82.09% 30.46% \$94 \$93,630,140.14 \$55,397,833 84.05% 37.10% \$82 \$77,216,778.81 \$45,686,594 85.81% 44.15% \$69 \$62,766,224.91 \$37,136,683 87.40% 51.45% \$52 \$50,043,754.25 \$29,609,221 88.82% 58.87% \$43 \$38,842,712.35 \$22,981,938 90.08% 66.28% \$33 \$28,981,157.70 \$17,147,185 91.22% 73.57% \$25 \$20,298,906.46 \$12,010,186 92.22% 80.66% \$16 \$12,654,930.49 \$7,487,501 93.12% 87.46% \$12 \$5,925,066.66 \$3,505,664 93.92% 93.92%
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 Posted: November 18, 2006, 8:43 am - IP Logged

MM has rolled over.  The cash jackpot is \$9.4M.  The advertised annuity value is \$16M.  Second round draws sell on average \$14.9M tickets.  If this is the number of tickets that sell, the probability of various numbers of winners is distributed as follows:

 0 91.85% 1 7.81% 2 0.33% 3 0.01%

Lotteryreport has not yet posted the sales figures and I will update the long term jackpot evolution figures when she has done so.

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 Posted: November 19, 2006, 12:41 am - IP Logged

The sales figures for the last MM drawing are now available.  Here is the long term probability model function data for the probability for the evolution of the jackpot:

 Historical Average Advertised Annuity Jackpots Model Calculated Annuity Prize Model Calculated Cash Value Single Draw Calculated Model Rollover Probability Overall Calculated Model Rollover Probability \$593,382,592.54 \$348,612,273 44.87% 0.21% \$517,163,807.92 \$303,833,737 49.38% 0.46% \$450,060,465.71 \$264,410,524 53.73% 0.94% \$390,982,397.77 \$229,702,159 57.88% 1.75% \$315 \$338,969,815.43 \$199,144,767 61.79% 3.02% \$262 \$293,177,716.68 \$172,241,909 65.45% 4.88% \$247 \$252,862,158.13 \$148,556,518 68.85% 7.46% \$217 \$217,368,168.89 \$127,703,799 72.00% 10.83% \$180 \$186,119,109.77 \$109,344,977 74.88% 15.04% \$153 \$158,607,305.18 \$93,181,792 77.52% 20.09% \$131 \$134,385,795.39 \$78,951,655 79.91% 25.92% \$111 \$113,061,075.13 \$66,423,382 82.09% 32.43% \$94 \$94,286,700.69 \$55,393,437 84.05% 39.51% \$82 \$77,757,661.57 \$45,682,626 85.81% 47.01% \$68 \$63,205,425.22 \$37,133,187 87.40% 54.78% \$56 \$50,393,574.43 \$29,606,225 88.82% 62.68% \$46 \$39,113,966.49 \$22,979,455 90.09% 70.57% \$34 \$29,183,351.68 \$17,145,219 91.22% 78.34% \$25 \$20,440,396.15 \$12,008,733 92.23% 85.88% \$16 \$12,743,060.91 \$7,486,548 93.12% 93.12% \$12 \$5,966,294.22 \$3,505,198 93.92% 100.00%
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 Posted: November 19, 2006, 12:51 am - IP Logged

The Powerball jackpot has rolled over.  The cash value is \$26.1M; the advertised annuity is \$54M.

On average, sixth drawings in a powerball jackpot run sell \$16.89M in tickets.

Were the average number of tickets to sell this time out, the randomized probability of various numbers of winners is as follows:

 0 89.08% 1 10.30% 2 0.60% 3 0.02%

The model function for the long term evolution of jackpots suggests the following:

 Average Historical Annuity Calculated Annuity Calculated Cash Value Projections Single Draw Rollover Probability Overall Probability from baseline \$563,775,073.13 \$272,491,285 27.11% 0.35% \$353 \$444,264,170.36 \$214,727,682 37.86% 1.29% \$300 \$355,325,804.62 \$171,740,806 48.54% 3.41% \$245 \$289,139,099.33 \$139,750,565 58.39% 7.03% \$208 \$239,883,866.00 \$115,943,869 67.01% 12.04% \$193 \$203,228,800.60 \$98,227,254 74.24% 17.97% \$180 \$175,950,606.43 \$85,042,793 80.11% 24.21% \$159 \$155,650,550.50 \$75,231,099 83.87% 30.22% \$132 \$139,545,043.85 \$67,446,771 84.50% 36.03% \$119 \$124,128,268.44 \$59,995,330 85.14% 42.64% \$105 \$109,400,224.25 \$52,876,775 85.78% 50.08% \$92 \$95,360,911.30 \$46,091,107 86.43% 58.38% \$80 \$82,010,329.57 \$39,638,326 87.08% 67.55% \$68 \$69,348,479.07 \$33,518,432 87.74% 77.57% \$57 \$57,375,359.80 \$27,731,424 88.40% 88.40%
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 Posted: November 22, 2006, 8:21 am - IP Logged

MM has rolled over.  The cash jackpot is \$13.6M.  The advertised annuity value is \$23M.  Second round draws sell on average \$15.9M tickets.  If this is the number of tickets that sell, the probability of various numbers of winners is distributed as follows:

 0 91.32% 1 8.29% 2 0.38% 3 0.01%

The long term probability model suggests the following:

 Historical Average Advertised Annuity Jackpots Model Calculated Annuity Prize Model Calculated Cash Value Single Draw Calculated Model Rollover Probability Overall Calculated Model Rollover Probability \$589,627,657.58 \$348,649,398 44.86% 0.22% \$513,867,975.02 \$303,852,368 49.37% 0.50% \$447,172,348.14 \$264,414,954 53.72% 1.01% \$388,456,333.32 \$229,695,919 57.87% 1.87% \$315 \$336,765,232.15 \$199,130,746 61.78% 3.24% \$262 \$291,258,568.45 \$172,222,458 65.45% 5.24% \$247 \$251,196,422.46 \$148,533,537 68.85% 8.01% \$217 \$215,927,400.07 \$127,678,810 71.99% 11.64% \$180 \$184,878,041.39 \$109,319,190 74.88% 16.16% \$153 \$157,543,496.54 \$93,156,154 77.52% 21.58% \$131 \$133,479,316.98 \$78,926,900 79.92% 27.84% \$111 \$112,294,228.99 \$66,400,066 82.09% 34.84% \$94 \$93,643,771.71 \$55,371,969 84.05% 42.44% \$82 \$77,224,696.31 \$45,663,299 85.82% 50.49% \$68 \$62,770,035.34 \$37,116,195 87.40% 58.84% \$56 \$50,044,761.93 \$29,591,685 88.82% 67.32% \$46 \$38,841,968.32 \$22,967,425 90.09% 75.79% \$34 \$28,979,501.67 \$17,135,705 91.22% 84.13% \$25 \$20,297,002.31 \$12,001,706 92.23% 92.23%
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 Posted: November 25, 2006, 5:14 am - IP Logged

MM has rolled over.  The cash jackpot is \$18.4M.  The advertised annuity value is \$31M.  Fourth round draws sell on average \$16.8M tickets.  If this is the number of tickets that sell, the probability of various numbers of winners is distributed as follows:

 0 90.90% 1 8.67% 2 0.41% 3 0.01%

The long term curve fitting model suggests the following for the probability for evolution of the jackpot.

 Historical Average Advertised Annuity Jackpots Model Calculated Annuity Prize Model Calculated Cash Value Single Draw Calculated Model Rollover Probability Overall Calculated Model Rollover Probability \$587,406,671.23 \$348,654,282 44.85% 0.24% \$511,928,935.92 \$303,854,594 49.37% 0.54% \$445,482,042.33 \$264,415,148 53.72% 1.09% \$386,985,458.53 \$229,694,595 57.87% 2.03% \$315 \$335,487,937.26 \$199,128,324 61.78% 3.51% \$262 \$290,152,047.12 \$172,219,280 65.44% 5.68% \$247 \$250,240,554.60 \$148,529,878 68.85% 8.69% \$217 \$215,104,435.51 \$127,674,891 71.99% 12.62% \$180 \$184,172,320.80 \$109,315,184 74.88% 17.52% \$153 \$156,941,205.21 \$93,152,199 77.52% 23.40% \$131 \$132,968,267.59 \$78,923,101 79.92% 30.19% \$111 \$111,863,669.95 \$66,396,501 82.09% 37.78% \$94 \$93,284,218.03 \$55,368,697 84.05% 46.02% \$82 \$76,927,780.47 \$45,660,360 85.82% 54.75% \$68 \$62,528,375.64 \$37,113,617 87.40% 63.80% \$56 \$49,851,846.38 \$29,589,483 88.82% 72.99% \$46 \$38,692,052.20 \$22,965,605 90.09% 82.18% \$34 \$28,867,517.14 \$17,134,268 91.22% 91.22%

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 Posted: November 26, 2006, 3:09 pm - IP Logged

Prob988, I hope MM continues to roll and get up to 365+ million.

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 Posted: November 29, 2006, 11:21 pm - IP Logged

MM has rolled over.  The cash jackpot is \$23.9M.  The advertised annuity value is \$40M.  Fifth round draws sell on average \$18.6M tickets.  If this is the number of tickets that sell, the probability of various numbers of winners is distributed as follows:

 0 89.94% 1 9.54% 2 0.51% 3 0.02%

The long term curve fitting model suggests the following for the probability for evolution of the jackpot:

 Historical Average Advertised Annuity Jackpots Model Calculated Annuity Prize Model Calculated Cash Value Single Draw Calculated Model Rollover Probability Overall Calculated Model Rollover Probability \$583,563,750.38 \$348,679,341 44.84% 0.26% \$508,557,733.71 \$303,863,246 49.36% 0.59% \$442,529,465.60 \$264,411,356 53.71% 1.20% \$384,404,366.01 \$229,681,609 57.86% 2.23% \$315 \$333,236,475.36 \$199,108,794 61.78% 3.85% \$262 \$288,193,059.48 \$172,195,353 65.44% 6.23% \$247 \$248,541,057.25 \$148,503,282 68.85% 9.53% \$217 \$213,635,150.36 \$127,647,002 71.99% 13.83% \$180 \$182,907,261.04 \$109,287,088 74.88% 19.22% \$153 \$155,857,306.86 \$93,124,741 77.52% 25.66% \$131 \$132,045,062.10 \$78,896,925 79.92% 33.10% \$111 \$111,082,993.32 \$66,372,089 82.09% 41.42% \$94 \$92,629,952.36 \$55,346,397 84.05% 50.46% \$82 \$76,385,624.37 \$45,640,411 85.82% 60.03% \$68 \$62,085,640.30 \$37,096,170 87.41% 69.95% \$56 \$49,497,274.41 \$29,574,621 88.83% 80.03% \$46 \$38,415,656.76 \$22,953,355 90.09% 90.09%
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 Posted: December 3, 2006, 10:12 am - IP Logged

The MM jackpot has been won.    The cash prize is reset to \$7.2M and the advertised annuity is \$12M.  Average 1st draws produce 15.2M in sales.  Selling an average number of tickets would result in the following distribution for the probability of various numbers of winners:

 0 91.72% 1 7.93% 2 0.34% 3 0.01%

Here is the long term probability, which ignores minimums, focusing on models generated from sales figures and what sales actually contribute to jackpots:

 Historical Average Advertised Annuity Jackpots Model Calculated Annuity Prize Model Calculated Cash Value Single Draw Calculated Model Rollover Probability Overall Calculated Model Rollover Probability \$589,340,718.90 \$348,693,259 44.83% 0.19% \$513,574,074.30 \$303,864,661 49.35% 0.43% \$446,878,967.66 \$264,403,389 53.70% 0.88% \$388,169,264.02 \$229,666,815 57.85% 1.64% \$315 \$336,488,871.33 \$199,089,249 61.77% 2.84% \$262 \$290,996,170.42 \$172,172,734 65.44% 4.59% \$247 \$250,950,309.16 \$148,478,933 68.85% 7.01% \$217 \$215,699,137.59 \$127,621,990 71.99% 10.19% \$180 \$184,668,587.67 \$109,262,248 74.88% 14.15% \$153 \$157,353,324.52 \$93,100,717 77.52% 18.90% \$131 \$133,308,516.95 \$78,874,206 79.92% 24.38% \$111 \$112,142,593.43 \$66,351,034 82.10% 30.50% \$94 \$93,510,865.27 \$55,327,262 84.06% 37.15% \$82 \$77,109,913.33 \$45,623,365 85.82% 44.20% \$68 \$62,672,646.84 \$37,081,316 87.41% 51.50% \$56 \$49,963,953.81 \$29,562,006 88.83% 58.92% \$46 \$38,776,872.20 \$22,942,983 90.10% 66.32% \$34 \$28,929,219.51 \$17,116,455 91.23% 73.61% \$25 \$20,260,625.97 \$11,987,537 92.24% 80.69% \$16 \$12,629,922.85 \$7,472,704 93.13% 87.48% \$12 \$5,912,843.58 \$3,498,432 93.93% 93.93%
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 Posted: December 6, 2006, 9:48 am - IP Logged

MM has rolled over.  The cash jackpot is \$9.0M.  The advertised annuity value is \$15M.  Second round draws sell on average \$14.9M tickets.  If this is the number of tickets that sell, the probability of various numbers of winners is distributed as follows:

 0 91.87% 1 7.79% 2 0.33% 3 0.01%

The long term modeling program I use, which does not account for jackpot minima, suggests the following:

 Historical Average Advertised Annuity Jackpots Model Calculated Annuity Prize Model Calculated Cash Value Single Draw Calculated Model Rollover Probability Overall Calculated Model Rollover Probability \$581,306,498.72 \$348,783,899 44.80% 0.21% \$506,522,262.92 \$303,913,358 49.32% 0.46% \$440,699,603.43 \$264,419,762 53.68% 0.94% \$382,764,632.81 \$229,658,780 57.84% 1.75% \$315 \$331,772,150.20 \$199,063,290 61.76% 3.02% \$262 \$286,890,220.41 \$172,134,132 65.43% 4.89% \$247 \$247,386,601.10 \$148,431,961 68.84% 7.47% \$217 \$212,616,796.26 \$127,570,078 71.99% 10.85% \$180 \$182,013,541.44 \$109,208,125 74.88% 15.08% \$153 \$155,077,548.85 \$93,046,529 77.52% 20.14% \$131 \$131,369,361.56 \$78,821,617 79.93% 25.97% \$111 \$110,502,183.84 \$66,301,310 82.10% 32.50% \$94 \$92,135,570.62 \$55,281,342 84.06% 39.58% \$82 \$75,969,873.19 \$45,581,924 85.83% 47.08% \$68 \$61,741,350.48 \$37,044,810 87.42% 54.86% \$56 \$49,217,866.15 \$29,530,720 88.84% 62.75% \$46 \$38,195,101.31 \$22,917,061 90.11% 70.64% \$34 \$28,493,221.10 \$17,095,933 91.24% 78.39% \$25 \$19,953,940.73 \$11,972,364 92.25% 85.92% \$16 \$12,437,943.02 \$7,462,766 93.14% 93.14%

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