Mega Millions
March 29th, 2012 at 5:03:18 PM
permalink
Wait a sec -
You're predicting a 10.9% chance of 5 or more winners, even though that's never happened before?
You're predicting a 10.9% chance of 5 or more winners, even though that's never happened before?
I invented a few casino games. Info:
http://www.DaveMillerGaming.com/ —————————————————————————————————————
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
March 29th, 2012 at 5:07:10 PM
permalink
Quote: DJTeddyBearWait a sec -
You're predicting a 10.9% chance of 5 or more winners, even though that's never happened before?
Sure - 440,000,000 sales for one drawing has never happened before either:)
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
March 29th, 2012 at 5:16:25 PM
permalink
Who wants to come with me to CA tomorrow to buy lottery tickets?
March 29th, 2012 at 5:51:53 PM
permalink
Wait it minute. You're saying that it's most likely that 2 people will win it? Well, then why can't one of those people be ME! Oh, wait ...Quote: JB
- There is about a 1 in 12 chance that nobody will win it
- The most likely outcome is a 2-way split
- A 3-way split is more likely than a single winner
- 5 or more winners is more likely than no winner
"Dice, verily, are armed with goads and driving-hooks, deceiving and tormenting, causing grievous woe." -Rig Veda 10.34.4
March 29th, 2012 at 6:07:26 PM
permalink
Quote: JB
- There is about a 1 in 12 chance that nobody will win it
- The most likely outcome is a 2-way split
- A 3-way split is more likely than a single winner
- 5 or more winners is more likely than no winner
Days are days since the previous win.
I posted this table earlier, but it may have gotten lost in the later posts. The top four awards have been awarded to multiple tickets.
| Winners | Date | Prize | Days |
|---|---|---|---|
| 2 | 6-Mar-2007 | $390 | 56 |
| 2 | 4-Jan-2011 | $380 | 56 |
| 2 | 28-Aug-2009 | $336 | 52 |
| 4 | 31-Aug-2007 | $330 | 56 |
| 1 | 25-Mar-2011 | $319 | 52 |
| 1 | 15-Nov-2005 | $315 | 60 |
| 1 | 2-Jul-2004 | $294 | 56 |
| 1 | 22-Feb-2008 | $275 | 52 |
| 1 | 28-Feb-2006 | $270 | 53 |
| 1 | 4-May-2010 | $266 | 53 |
| 1 | 18-Apr-2006 | $265 | 49 |
| 1 | 16-Sep-2005 | $258 | 56 |
But some of the small jackpots have also been split among 2 winners.
| Winners | Date | Prize | Days |
|---|---|---|---|
| 2 | 6-Mar-07 | $390 | 56 |
| 2 | 4-Jan-11 | $380 | 56 |
| 2 | 28-Aug-09 | $336 | 52 |
| 4 | 31-Aug-07 | $330 | 56 |
| 3 | 1-May-09 | $227 | 49 |
| 2 | 18-Dec-07 | $163 | 46 |
| 2 | 30-Sep-11 | $113 | 42 |
| 2 | 1-Feb-11 | $93 | 28 |
| 2 | 10-Nov-09 | $77 | 25 |
| 3 | 11-Nov-03 | $70 | 35 |
| 2 | 15-May-09 | $38 | 14 |
| 2 | 29-Nov-05 | $35 | 14 |
| 2 | 23-May-08 | $17 | 7 |
The three way split for $70 million was 2/3 and 1/3 as one of the winners had bet the same set of numbers twice. Had that person only bet one ticket they would have won $35 million instead of $46.67 million, so that single presumed mistake gave them an extra $11.67 million.
No hard statistics are done, because I don't know how many tickets were purchased for each win.
The $17 million prize split two ways, is the lowest amount of money won by a jackpot winner. One time a single jackpot winner won only $10 million, and seven times a single jackpot winner won only $12 million.
March 29th, 2012 at 6:19:45 PM
permalink
What is the probability that, among 440,000,000 Mega Millions tickets, all 175,711,536 possible combinations are represented at least once?
March 29th, 2012 at 7:05:11 PM
permalink
Quote: JBWhat is the probability that, among 440,000,000 Mega Millions tickets, all 175,711,536 possible combinations are represented at least once?
That's an interesting question. With dice you set up a Markov matrix, which you can't do with all those possibilities. So there must be an approximation.
With dice if you roll them 12 times there is a 43.78% probability that all six sides will turn up so there is 56.22% chance that one die won't be covered. So I am guessing it is about 50%.
March 29th, 2012 at 7:53:11 PM
permalink
Trying to add images. No use.
The EV of the jackpot prize starts to decrease when the last un-won jackpot was 320 million. So, we're at a sweet spot right now.
The EV of the jackpot prize starts to decrease when the last un-won jackpot was 320 million. So, we're at a sweet spot right now.
I heart Crystal Math.
March 29th, 2012 at 8:04:16 PM
permalink
Quote: CrystalMathThe EV of the jackpot prize starts to decrease when the last un-won jackpot was 320 million. So, we're at a sweet spot right now.
Isn't that simply the observation that the biggest prize won by a single ticket was $319 million (last year)?
The expected outcome is that two people will win $270 million apiece.
| Winners | Date | Prize | Days |
|---|---|---|---|
| 2 | 6-Mar-2007 | $390 | 56 |
| 2 | 4-Jan-2011 | $380 | 56 |
| 2 | 28-Aug-2009 | $336 | 52 |
| 4 | 31-Aug-2007 | $330 | 56 |
| 1 | 25-Mar-2011 | $319 | 52 |
| 1 | 15-Nov-2005 | $315 | 60 |
| 1 | 2-Jul-2004 | $294 | 56 |
| 1 | 22-Feb-2008 | $275 | 52 |
| 1 | 28-Feb-2006 | $270 | 53 |
| 1 | 4-May-2010 | $266 | 53 |
| 1 | 18-Apr-2006 | $265 | 49 |
| 1 | 16-Sep-2005 | $258 | 56 |
March 29th, 2012 at 8:10:07 PM
permalink
Quote: pacomartinIsn't that simply the observation that the biggest prize won by a single ticket was $319 million (last year)?
The expected outcome is that two people will win $270 million apiece.
DAMN !!! Now I have got to buy the wife a ticket too . Don't want to split the jackpot with a stranger.
Reading this just cost me another dollar.
March 29th, 2012 at 8:10:35 PM
permalink
I was considering getting 5 quick picks, then buying one that's the same numbers as the last drawing. I just think that would be an awesome story if it hit again. Win the biggest jackpot ever, and it happened to be the same numbers as the last drawing, wow.. It's gotta happen someday right?
March 29th, 2012 at 8:15:15 PM
permalink
Quote: pacomartinThe expected outcome is that two people will win $270 million apiece.
...unless you compare the probability of a 2-way split (25.63%) to the probability of something other than a 2-way split (74.37%)...
(I'm just teasing; I'm not trying to incite an argument.)
March 29th, 2012 at 8:16:38 PM
permalink
The first image shows ticket sales as a function of the last prize not won. This is how I estimated the 341 million in ticket sales.
The second image shows the ev of the next jackpot as a function of the last prize not won. The ev is caculated by taking into account the probability of having other winners. It also assumes that the tickets purchased are randomly distributed. The ev shown is for the Jackpot amount, not the cash amount. To get the cash amount, multiply by 0.718. This makes the maximum EV 0.95. This EV does not take into account the other prizes.
The second image shows the ev of the next jackpot as a function of the last prize not won. The ev is caculated by taking into account the probability of having other winners. It also assumes that the tickets purchased are randomly distributed. The ev shown is for the Jackpot amount, not the cash amount. To get the cash amount, multiply by 0.718. This makes the maximum EV 0.95. This EV does not take into account the other prizes.
I heart Crystal Math.
March 29th, 2012 at 8:18:07 PM
permalink
Yeah, I didn't have time to analyze each week's draw, Wizard, so I used evidence from the past three big wins which showed marginally an increase in EV as the increase in tickets sold didn't decrease the odds of more prizes being drawn as much. While three data points isn't enough, it does show a very very basic trend.
No one else's come up with the desired chart: Tickets sold vs EV of the prize. You can estimate tickets sold by multiplying the total # of prizes for the draw (available at Megamillions x 39.89. You can then estimate the EV of your ticket by taking $.182 (which is the basic EV of all of the nonwinning prizes and adding the following)
Sum of (w from 0 to 100): COMBIN(n,w)*(1/175711536)^w*((175711535)/175711536)^(n-w)/w
I use 100 as for this draw the odds of 100 prizes being won is fairly miniscule.
So for example, for a draw with 200,000,000 tickets the EV is .501499. Multiply that EV by the prize value V (where v is the after tax value) and divie by 175711536 to get the expected value of a ticket, and add the .182.
I bought my 10 tickets at the Chevron in San Diego today on the way to the airport. I'm keeping all of the prize for myself when I win. Screw... never mind.
No one else's come up with the desired chart: Tickets sold vs EV of the prize. You can estimate tickets sold by multiplying the total # of prizes for the draw (available at Megamillions x 39.89. You can then estimate the EV of your ticket by taking $.182 (which is the basic EV of all of the nonwinning prizes and adding the following)
Sum of (w from 0 to 100): COMBIN(n,w)*(1/175711536)^w*((175711535)/175711536)^(n-w)/w
I use 100 as for this draw the odds of 100 prizes being won is fairly miniscule.
So for example, for a draw with 200,000,000 tickets the EV is .501499. Multiply that EV by the prize value V (where v is the after tax value) and divie by 175711536 to get the expected value of a ticket, and add the .182.
I bought my 10 tickets at the Chevron in San Diego today on the way to the airport. I'm keeping all of the prize for myself when I win. Screw... never mind.
-----
You want the truth! You can't handle the truth!
March 29th, 2012 at 8:27:00 PM
permalink
Quote: CrystalMathThe first image shows ticket sales as a function of the last prize not won. This is how I estimated the 341 million in ticket sales.
The second image shows the ev of the next jackpot as a function of the last prize not won. The ev is caculated by taking into account the probability of having other winners. It also assumes that the tickets purchased are randomly distributed. The ev shown is for the Jackpot amount, not the cash amount. To get the cash amount, multiply by 0.718. This makes the maximum EV 0.95. This EV does not take into account the other prizes.
it looks like the regression breaks down a bit at the end. 9 of the last 11 data points are above the trend line. The trend fits the lower jackpot data almost perfectly, but seems to under-estimate sales at higher jackpot levels.
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
March 29th, 2012 at 8:32:20 PM
permalink
Quote: rdw4potusit looks like the regression breaks down a bit at the end. 9 of the last 11 data points are above the trend line. The trend fits the lower jackpot data almost perfectly, but seems to under-estimate sales at higher jackpot levels.
I agree. That's also why I previously said I thought the sales could go to 500 million.
If it does go to 500 million in sales, the EV of this jackpot should be 1.08, or 0.78 for the cash prize.
I heart Crystal Math.
March 29th, 2012 at 8:33:41 PM
permalink
Quote: CrystalMathI agree. That's also why I previously said I thought the sales could go to 500 million.
If it does go to 500 million in sales, the EV of this jackpot should be 1.08, or 0.78 for the cash prize.
Is that after the adjustment for the increase in the jackpot due to the increased sales?
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
March 29th, 2012 at 8:47:54 PM
permalink
Quote: rdw4potusIs that after the adjustment for the increase in the jackpot due to the increased sales?
I'm not taking into account the estimated jackpot amount from the lottery. But, I know that plays into the psychology.
If I worked for the lottery, I would have started with 510 million and estimated 341 million in sales. The benefit they have is that they know the number of tickets already sold and are forecasting sales through tomorrow.
I heart Crystal Math.
March 29th, 2012 at 10:31:08 PM
permalink
Any thoughts on how high the jackpot would go if nobody wins it on Friday night? Would we see $750-800 Million for the drawing on Tuesday, 4/3?
"Bite my Glorious Golden Ass!" - Bender Bending Rodriguez
March 30th, 2012 at 12:25:11 AM
permalink
Quote: pacomartinQuote: JBWhat is the probability that, among 440,000,000 Mega Millions tickets, all 175,711,536 possible combinations are represented at least once?
That's an interesting question. With dice you set up a Markov matrix, which you can't do with all those possibilities. So there must be an approximation.
With dice if you roll them 12 times there is a 43.78% probability that all six sides will turn up so there is 56.22% chance that one die won't be covered. So I am guessing it is about 50%.
A simulation I've been running seems to indicate that the probability is close to 0. After 700 trials so far, I have not encountered any successes.
Each trial is of course rather slow: 440,000,000 iterations to pick the ticket and up to 175,711,536 iterations to determine success or failure. 3 of my CPU's 4 cores are running their own simulator; the 4th one is monitoring their results. But after so many trials with no successes, I doubt I will keep it running for much longer.
March 30th, 2012 at 12:27:58 AM
permalink
Bought $5 worth of quick picks tonight for what it's worth. Never know. Thanks for all of the info in this thread though, very interesting. For the record none of my megaball numbers repeated and only two regular numbers repeated once on my quick picks.
March 30th, 2012 at 1:16:12 AM
permalink
Quote: JBA simulation I've been running seems to indicate that the probability is close to 0. After 700 trials so far, I have not encountered any successes.
Each trial is of course rather slow: 440,000,000 iterations to pick the ticket and up to 175,711,536 iterations to determine success or failure. 3 of my CPU's 4 cores are running their own simulator; the 4th one is monitoring their results. But after so many trials with no successes, I doubt I will keep it running for much longer.
But this implies a random pick of a combination. In reality it doesnt happen. So i think its pretty sure 0.
March 30th, 2012 at 1:25:34 AM
permalink
Quote: edwardBut this implies a random pick of a combination. In reality it doesnt happen. So i think its pretty sure 0.
Correct - it would only be accurate if everyone who played always bought quick picks, and never chose their own numbers. Regardless, it would still be a good estimate.
March 30th, 2012 at 3:44:20 AM
permalink
So, here's my table. It shows all drawings from 2007 onward where the estimated jackpot has been for 150M+. I don't care who won. I'm looking at the expected value of buying a ticket based on the estimated jackpot.
I assume that the tickets sold is the total # of prizes times 39.89. I assume that taxes are 39.6%. I add .182 to the expected value for all non winning prizes. The EV appears to increase with the jackpot size despite tickets sold.
I assume that the tickets sold is the total # of prizes times 39.89. I assume that taxes are 39.6%. I add .182 to the expected value for all non winning prizes. The EV appears to increase with the jackpot size despite tickets sold.
| Date | Prizes | Est Tickets | Est Jackpot | Jackpot EV | Less 39.6% tax | EV $1 ticket |
|---|---|---|---|---|---|---|
| 3/6/07 | 5,331,731 | 212,682,750 | $390,000,000 | $282,000,111 | $170,328,067 | 1.1514 |
| 1/4/11 | 5,515,767 | 220,023,946 | $380,000,000 | $271,546,192 | $164,013,900 | 1.1154 |
| 3/27/12 | 4,715,569 | 188,104,047 | $363,000,000 | $272,994,123 | $164,888,450 | 1.1204 |
| 8/28/09 | 3,732,067 | 148,872,153 | $333,000,000 | $266,427,080 | $160,921,956 | 1.0978 |
| 8/31/07 | 3,547,787 | 141,521,223 | $330,000,000 | $267,074,833 | $161,313,199 | 1.1001 |
| 3/25/11 | 3,587,257 | 143,095,682 | $319,000,000 | $257,539,305 | $155,553,740 | 1.0673 |
| 3/23/12 | 2,879,006 | 114,843,549 | $290,000,000 | $244,597,823 | $147,737,085 | 1.0228 |
| 2/22/08 | 2,766,281 | 110,346,949 | $275,000,000 | $233,552,879 | $141,065,939 | 0.9848 |
| 3/2/07 | 2,624,669 | 104,698,046 | $275,000,000 | $235,581,237 | $142,291,067 | 0.9918 |
| 5/4/10 | 2,041,460 | 81,433,839 | $266,000,000 | $236,061,777 | $142,581,313 | 0.9935 |
| 8/25/09 | 2,472,384 | 98,623,398 | $252,000,000 | $217,888,095 | $131,604,409 | 0.9310 |
| 8/28/07 | 2,114,019 | 84,328,218 | $250,000,000 | $220,895,954 | $133,421,156 | 0.9413 |
| 3/22/11 | 1,918,524 | 76,529,922 | $244,000,000 | $218,141,802 | $131,757,648 | 0.9319 |
| 12/31/10 | 3,162,105 | 126,136,368 | $242,000,000 | $200,589,556 | $121,156,092 | 0.8715 |
| 3/20/12 | 2,024,644 | 80,763,049 | $241,000,000 | $214,091,817 | $129,311,457 | 0.9179 |
| 5/1/09 | 1,953,257 | 77,915,422 | $225,000,000 | $200,736,835 | $121,245,048 | 0.8720 |
| 4/30/10 | 1,610,768 | 64,253,536 | $224,000,000 | $203,974,854 | $123,200,812 | 0.8832 |
| 2/19/08 | 1,809,824 | 72,193,879 | $220,000,000 | $197,969,079 | $119,573,324 | 0.8625 |
| 2/27/07 | 1,899,348 | 75,764,992 | $216,000,000 | $193,331,161 | $116,772,021 | 0.8466 |
| 3/3/09 | 2,068,570 | 82,515,257 | $212,000,000 | $187,832,957 | $113,451,106 | 0.8277 |
| 8/21/09 | 1,964,485 | 78,363,307 | $207,000,000 | $184,553,520 | $111,470,326 | 0.8164 |
| 12/12/08 | 2,039,073 | 81,338,622 | $207,000,000 | $183,728,573 | $110,972,058 | 0.8136 |
| 12/27/11 | 1,754,475 | 69,986,008 | $206,000,000 | $185,984,889 | $112,334,873 | 0.8213 |
| 8/24/07 | 1,862,997 | 74,314,950 | $206,000,000 | $184,782,428 | $111,608,587 | 0.8172 |
| 3/18/11 | 1,567,685 | 62,534,955 | $201,000,000 | $183,500,243 | $110,834,147 | 0.8128 |
| 3/16/12 | 1,796,988 | 71,681,851 | $200,000,000 | $180,110,001 | $108,786,440 | 0.8011 |
| 12/28/10 | 1,545,752 | 61,660,047 | $200,000,000 | $182,825,257 | $110,426,455 | 0.8105 |
| 10/16/09 | 1,473,181 | 58,765,190 | $200,000,000 | $183,613,791 | $110,902,730 | 0.8132 |
| 5/16/08 | 1,523,173 | 60,759,371 | $196,000,000 | $179,408,982 | $108,363,025 | 0.7987 |
| 4/27/10 | 1,246,496 | 49,722,725 | $191,000,000 | $177,714,796 | $107,339,737 | 0.7929 |
| 4/28/09 | 1,411,462 | 56,303,219 | $181,000,000 | $166,778,717 | $100,734,345 | 0.7553 |
| 2/15/08 | 1,470,640 | 58,663,830 | $179,000,000 | $164,359,083 | $99,272,886 | 0.7470 |
| 2/23/07 | 1,296,058 | 51,699,754 | $177,000,000 | $164,208,329 | $99,181,831 | 0.7465 |
| 12/23/11 | 1,481,015 | 59,077,688 | $173,000,000 | $158,752,220 | $95,886,341 | 0.7277 |
| 3/15/11 | 1,171,381 | 46,726,388 | $172,000,000 | $160,745,040 | $97,090,004 | 0.7346 |
| 3/13/12 | 1,211,678 | 48,333,835 | $171,000,000 | $159,432,295 | $96,297,106 | 0.730 |
| 2/27/09 | 1,477,970 | 58,956,223 | $171,000,000 | $156,945,251 | $94,794,932 | 0.7215 |
| 8/21/07 | 1,156,584 | 46,136,136 | $171,000,000 | $159,949,460 | $96,609,474 | 0.7318 |
| 10/13/09 | 1,104,185 | 44,045,940 | $170,000,000 | $159,503,888 | $96,340,349 | 0.7303 |
| 8/18/09 | 1,371,730 | 54,718,310 | $170,000,000 | $157,011,354 | $94,834,858 | 0.7217 |
| 12/9/08 | 1,469,204 | 58,606,548 | $170,000,000 | $156,108,498 | $94,289,533 | 0.7186 |
| 12/24/10 | 1,377,480 | 54,947,677 | $168,000,000 | $155,111,453 | $93,687,317 | 0.7152 |
| 4/23/10 | 1,126,422 | 44,932,974 | $166,000,000 | $155,547,790 | $93,950,865 | 0.7167 |
| 5/13/08 | 1,142,357 | 45,568,621 | $166,000,000 | $155,402,358 | $93,863,024 | 0.7162 |
| 12/18/07 | 1,103,740 | 44,028,189 | $163,000,000 | $152,940,073 | $92,375,804 | 0.7077 |
| 12/22/09 | 1,169,746 | 46,661,168 | $162,000,000 | $151,413,942 | $91,454,021 | 0.7025 |
| 4/24/09 | 1,357,303 | 54,142,817 | $155,000,000 | $143,279,470 | $86,540,800 | 0.6745 |
| 12/20/11 | 1,113,015 | 44,398,168 | $152,000,000 | $142,541,393 | $86,095,001 | 0.6720 |
| 3/11/11 | 1,129,667 | 45,062,417 | $151,000,000 | $141,465,322 | $85,445,054 | 0.6683 |
-----
You want the truth! You can't handle the truth!
March 30th, 2012 at 5:00:59 AM
permalink
You're also not taking into account how the news media works.Quote: CrystalMathI'm not taking into account the estimated jackpot amount from the lottery. But, I know that plays into the psychology.
If I worked for the lottery, I would have started with 510 million and estimated 341 million in sales.
Sure, they are all going to be talking about the jackpot. That's free publicity.
But if you pepper the mix by revising the estimates upward several times, you get more of that free publicity.
I invented a few casino games. Info:
http://www.DaveMillerGaming.com/ —————————————————————————————————————
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
March 30th, 2012 at 6:03:32 AM
permalink
Quote: JBWhat is the probability that, among 440,000,000 Mega Millions tickets, all 175,711,536 possible combinations are represented at least once?
I know you've already investigated some more, but I initially thought it was 0 (a "soft" 0) You calculated that the chance of 0 winners is about 8%. That means that 8% of 175 million tickets will be unrepresented, which is enormous. If we really wanted to estimate a probability, we could calculate a standard deviation on the number of unchosen tickets.
I heart Crystal Math.
March 30th, 2012 at 6:20:21 AM
permalink
Quote: CrystalMathI know you've already investigated some more, but I initially thought it was 0 (a "soft" 0) You calculated that the chance of 0 winners is about 8%. That means that 8% of 175 million tickets will be unrepresented, which is enormous. If we really wanted to estimate a probability, we could calculate a standard deviation on the number of unchosen tickets.
It's probably 0.00000000000000000......1, which is the missing link (and bone of contention) in this thread.
March 30th, 2012 at 6:23:28 AM
permalink
On this morning's news they showed the lines at Primm (stateline) from yesterday afternoon. Most people were waiting 4 to 6 hours to get a ticket. Then they showed the store close at 8:00PM. Talk about a bunch of pissed off people who waited and didn't get tickets.
March 30th, 2012 at 7:24:09 AM
permalink
Quote: thlfOn this morning's news they showed the lines at Primm (stateline) from yesterday afternoon. Most people were waiting 4 to 6 hours to get a ticket. Then they showed the store close at 8:00PM. Talk about a bunch of pissed off people who waited and didn't get tickets.
Unless there's a law that forbids the sale of tickets after 8 pm, the owner has to be incredibly stupid to close when there's a long line of eager customers waiting.
Donald Trump is a fucking criminal
March 30th, 2012 at 7:30:14 AM
permalink
Quote: Toes14Any thoughts on how high the jackpot would go if nobody wins it on Friday night? Would we see $750-800 Million for the drawing on Tuesday, 4/3?
I think 1.2 billion if it doesn't hit today. They won't be able to post it on the billboards if that happens.
I heart Crystal Math.
March 30th, 2012 at 7:34:20 AM
permalink
Serves those people right for waiting rather than spending an extra, what? half hour driving to a place with short lines....Quote: thlfOn this morning's news they showed the lines at Primm (stateline) from yesterday afternoon. Most people were waiting 4 to 6 hours to get a ticket. Then they showed the store close at 8:00PM. Talk about a bunch of pissed off people who waited and didn't get tickets.
I invented a few casino games. Info:
http://www.DaveMillerGaming.com/ —————————————————————————————————————
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
March 30th, 2012 at 7:34:56 AM
permalink
Quote: CrystalMathI think 1.2 billion if it doesn't hit today. They won't be able to post it on the billboards if that happens.
I remember when the Powerball jackpot hit 100 million for the first time back in the 90s - they had to paint a '1' in front of all the digital billboards that only had 2 digits.
"So drink gamble eat f***, because one day you will be dust." -ontariodealer
March 30th, 2012 at 8:03:20 AM
permalink
Quote: CrystalMathThe first image shows ticket sales as a function of the last prize not won...
Good stuff CM. I'm hoping to find the time today to do the same type of analysis. What this shows is that this is not a good time to play, due to the induced demand for tickets, and thus jackpot sharing.
Regarding the number of winners today, the Poisson distribution is perfect for estimating that. First, let's estimate the number of tickets sold. Here is a good source for that: http://lottoreport.com/mmsales.htm. On the Mar 23 drawing the jackpot was 290M. On the Mar 27 drawing it was 363M, a growth of 73M. Between the 3/23 and 3/27 drawings 190.922 million tickets were sold. So, 38.24% of ticket sales went into the jackpot meter. In other words, for every dollar in growth, 2.6154 tickets were sold.
The MegaMillions web site is estimating a jackpot of 540M for today. That is a growth of 177M since the the March 27 draw. At 2.6154 tickets per dollar in growth, that would be 463M tickets sold. The chances of winning are 1 in 175,711,536. So the expected number of winners is 463/175.711 = 2.6346.
Using the Poisson distribution, the probability of n winners is exp(-n)*2.6346^n/n!.
| Winners | Probability |
|---|---|
| 0 | 7.17% |
| 1 | 18.90% |
| 2 | 24.90% |
| 3 | 21.87% |
| 4 | 14.40% |
| 5 | 7.59% |
| 6 | 3.33% |
| 7 | 1.25% |
| 8 | 0.41% |
| 9 | 0.12% |
| 10 | 0.03% |
The average jackpot after sharing is 540/2.6346 = 205M. Not bad, but as CM showed, your expected win is greater with a smaller jackpot. I hope to work on an optimal jackpot size myself later today.
By the way, for my fellow Las Vegans, rather than wait hours at the Primm gas station for tickets, I heard there is a convenience store just past the CA/NV border on Nipton Road. It should be just about a 20 minute drive from Primm. You could also just go all the way to Baker, but I wouldn't be surprised if lots of people are doing that if they see a gigantic line in Primm.
p.s. Based on my initial results, the optimal jackpot size to buy a ticket is 340 million. My formula for expected ticket sales is 14.718×exp(0.0072*j), where j is the jackpot size, in millions.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 30th, 2012 at 9:19:58 AM
permalink
The jackpot has just been raised to $640M! THey said they expect to sell 1.2B tickets. (Mentioned on CNBC)
March 30th, 2012 at 9:20:21 AM
permalink
Jumped again. 640/462.
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
March 30th, 2012 at 9:22:01 AM
permalink
Based on the law of diminishing returns, what is the "optimal" number of tickets to buy? My friend and I were having a discussion about it, trying to figure out whether it's $1, $2, $100, etc.
Any thoughts?
Any thoughts?
March 30th, 2012 at 9:24:32 AM
permalink
Quote: WizardI heard there is a convenience store just past the CA/NV border on Nipton Road.
Not sure about Nipton Rd, but there's a great Shell station at the Cima Rd exit (one exit farther from NV).
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
March 30th, 2012 at 10:01:06 AM
permalink
Quote: slytherThe jackpot has just been raised to $640M! THey said they expect to sell 1.2B tickets. (Mentioned on CNBC)
That doesn't add up. Assuming we believe the 640M, that would mean a jackpot increase of 277M since the last draw. There are about 2.6154 tickets sold per dollar increase, which would mean 724M tickets sold.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 30th, 2012 at 10:25:45 AM
permalink
Quote: WizardThat doesn't add up. Assuming we believe the 640M, that would mean a jackpot increase of 277M since the last draw. There are about 2.6154 tickets sold per dollar increase, which would mean 724M tickets sold.
I agree, it doesn't add up. My estimates are slightly different, though. I think that 31% of sales go into the cash value of the jackpot. Based on that, I think that it takes 3.226 tickets to add $1 to the CASH VALUE jackpot prize (or 2.316 tickets to add $1 to the advertised jackpot).
Before today's drawing, 840 million tickets have been sold that have contributed to this jackpot and the estimated cash value is 462 million. Therefore, the estimated tickets to be sold for this drawing are 462,000,000 / .31 - 840,000,000 = 650,000,000.
I heart Crystal Math.
March 30th, 2012 at 10:28:13 AM
permalink
deleted
DUHHIIIIIIIII HEARD THAT!
March 30th, 2012 at 10:37:04 AM
permalink
With an estimated 650M in sales and a cash value of 462M, the expected value of a win, given a win, is 121.8 million for the cash option. The EV is 69% before taxes.
Based on this, it is better to play when the previous un-won jackpot is 180 million to about 320 million.
Based on this, it is better to play when the previous un-won jackpot is 180 million to about 320 million.
I heart Crystal Math.
March 30th, 2012 at 10:40:35 AM
permalink
deleted
DUHHIIIIIIIII HEARD THAT!
March 30th, 2012 at 10:45:29 AM
permalink
31.8 cents per $1 ticket is added to the cash value.
“Man Babes” #AxelFabulous
March 30th, 2012 at 10:54:30 AM
permalink
Quote: CrystalMathI agree, it doesn't add up. My estimates are slightly different, though. I think that 31% of sales go into the cash value of the jackpot. Based on that, I think that it takes 3.226 tickets to add $1 to the CASH VALUE jackpot prize (or 2.316 tickets to add $1 to the advertised jackpot).
A problem seems to be that the ratio of meter increase to tickets sold is not consistent from drawing to drawing.
| Date | Sales (mil) | Jackpot (mil) | Jackpot increase (mil) | Ratio sales to increase | Contribution rate |
|---|---|---|---|---|---|
| 3/27/2012 | $190.92 | $363.00 | $73.00 | 2.6154 | 38.24% |
| 3/23/2012 | $111.28 | $290.00 | $49.00 | 2.2711 | 44.03% |
| 3/20/2012 | $78.42 | $241.00 | $41.00 | 1.9128 | 52.28% |
| 3/16/2012 | $66.66 | $200.00 |
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 30th, 2012 at 11:02:56 AM
permalink
Quote: miplet31.8 cents per $1 ticket is added to the cash value.
Care to elaborate? Where that has merit is Paco says that 50% of money bet goes to prizes, and the value of the fixed prizes is 18.2%, leaving 31.8% to the jackpot. However, how do you refute the fact that the meter goes up faster than that, based on published tickets sold at http://lottoreport.com/mmsales.htm.
One possible explanation is that site includes Megaplier sales. Another is that if the lump sum option is invoked, prizes are much less than the stated jackpot.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 30th, 2012 at 11:10:15 AM
permalink
Quote: WizardA problem seems to be that the ratio of meter increase to tickets sold is not consistent from drawing to drawing.
Date Sales (mil) Jackpot (mil) Jackpot increase (mil) Ratio sales to increase Contribution rate 3/27/2012 $190.92 $363.00 $73.00 2.6154 38.24% 3/23/2012 $111.28 $290.00 $49.00 2.2711 44.03% 3/20/2012 $78.42 $241.00 $41.00 1.9128 52.28% 3/16/2012 $66.66 $200.00
I based mine only on the jackpots that were won and the total sales that went into it. I did not use un-won jackpots since those are less accurate estimates because the lottery won't go back and revise those numbers based on actual sales.
Also, I only looked at the last 6 jackpots because it looks like the cash value calculations use a different interest rate before that.
I heart Crystal Math.
March 30th, 2012 at 11:13:28 AM
permalink
http://apps.leg.wa.gov/WAC/default.aspx?cite=315-38&full=true#315-38-080
Ok I gotta head to bed now. :+)
Ok I gotta head to bed now. :+)
“Man Babes” #AxelFabulous
March 30th, 2012 at 11:21:02 AM
permalink
Here's what I based mine on:
| date | jackpot | estimated cash value (jp * 0.718) | total sales | contribution (cash value/sales) |
|---|---|---|---|---|
| 8/5/2011 | 99000000 | 71082000 | 225665521 | 0.314988305 |
| 8/19/2011 | 32000000 | 22976000 | 71556285 | 0.321089894 |
| 9/30/2011 | 113000000 | 81134000 | 266207219 | 0.30477761 |
| 11/1/2011 | 78000000 | 56004000 | 175037338 | 0.319954592 |
| 12/27/2011 | 206000000 | 147908000 | 483269026 | 0.306057273 |
| 1/24/2012 | 71000000 | 50978000 | 164658431 | 0.30959848 |
I heart Crystal Math.
March 30th, 2012 at 11:24:22 AM
permalink
I think one complicating factor might be the rules under which the lottery seeds the initial jackpot. I cannot find the printed rules right now, but I think they "gamble" a little and guarantee a jackpot that might not be fully funded for the next drawing after a winning draw. I think the plan is to hold out some amount from subsequent draws to cover the possibility of having to pay out these jackpots that are not fully funded by sales for that drawing. Once the prize has rolled over several times, there is less need to keep holding out more as a reserve, so I would expect the cash-value jackpot to increase at an increasing rate.
March 30th, 2012 at 11:32:21 AM
permalink
I also calculated that they discount the annuity payments at 2.875%. Based on that, over 26 payments, the cash value would be 0.7176 of the jackpot amount.
I heart Crystal Math.
