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What are the world records for:

1. Consecutive successful passes?

2. Consecutive successful point conversions?

3. Consecutive Come Out natural winners (7 & 11)?

4. Consecutive Come Out craps?

5. Consecutive rolls w/o hitting a 6? (not point conversion, just the number)

6. Consecutive rolls w/o hitting an 8? (not point conversion, just the number)

Are such records kept??

tuttigym

The Borgata says they only lost $185K on the table during the roll. It was full of small betters.

It is hard to conceive that that might be the current world record after all this time. It would seem that with all the current technology and improvements in communications, such records or numbers would be available even with all the privacy concerns. Or perhaps not.

tuttigym

Quote:tuttigymIt is hard to conceive that that might be the current world record after all this time. It would seem that with all the current technology and improvements in communications, such records or numbers would be available even with all the privacy concerns. Or perhaps not.

As far as I know there's no way to measure and account for every individual bet at a craps table. Sure the dealers keep track of individual bets, but not a list of all individual bets over time. Once they collect a bet or pay it off, that's it.

You could measure and track all the action in two different ways: 1) with a Rapid Craps table or a similar semi-electronic deal; 2) by using RFID enabled chips and recording the result of each roll in the same system tracking the chips. With that info, assuming you only track pass line bets, you could have accurate results for any number of rolls you wanted, or any time period you chose. Tracking all other bets would require more ifnormation input, such as which player bet which hardway, hop, etc.

Quote:NareedAs far as I know there's no way to measure and account for every individual bet at a craps table. Sure the dealers keep track of individual bets, but not a list of all individual bets over time. Once they collect a bet or pay it off, that's it.

You could measure and track all the action in two different ways: 1) with a Rapid Craps table or a similar semi-electronic deal; 2) by using RFID enabled chips and recording the result of each roll in the same system tracking the chips. With that info, assuming you only track pass line bets, you could have accurate results for any number of rolls you wanted, or any time period you chose. Tracking all other bets would require more ifnormation input, such as which player bet which hardway, hop, etc.

Score one (1!) for RapidCraps...finally.

Oh well just a thought.

Thanks all for jumping in.

tuttigym

I know at the Freemont Casino downtown they have an electronic display that the box keeps rolls of each shooters hand. Someone should keep track of records and have bonus payouts when someone breaks a record!Quote:rudeboyoii think it was four queens downtown las vegas where they had a clicker on the table that the boxman would hit after each roll a 7 wasnt rolled.

The problem is, the boxman frequently forgot to click, then would remember and estimate the number of rolls he missed. It was a joke.

Yeah, that roll at Borgata had to go to the video to get the accurate count.

Anyway, it was not uncommon toward the end of the month to see the numbers be in the high 90s to low 100s. Not sure what the record was for that promotion, but 154 seems like a low number to me to be the world record.

Quote:7winnerI know at the Freemont Casino downtown they have an electronic display that the box keeps rolls of each shooters hand. Someone should keep track of records and have bonus payouts when someone breaks a record!

The Fremont runs a promo called the "Sharpshooter Roll-athon," for which they need to keep track of all shots made. See The Wizard's review for more info.

Quote:Tiltpoul

Anyway, it was not uncommon toward the end of the month to see the numbers be in the high 90s to low 100s. Not sure what the record was for that promotion, but 154 seems like a low number to me to be the world record.

That 154 roll was touted as a "world record" not yet challenged as inaccurate. Apparently the documentation is irrefutable.

The Wizard has written and stated that rolling the dice 75+ times w/o a 7 out is a 47,619 to 1 shot. That is where his calculations end. Doubling that number at 154 would be astronomical I guess.

Is that "high 90s to low 100s" any given month or consecutive months, and if so, how many months???

Thanks for the links, I am learning some stuff here and have some questions down the road.

tuttigym

Quote:7winnerI know at the Freemont Casino downtown they have an electronic display that the box keeps rolls of each shooters hand. Someone should keep track of records and have bonus payouts when someone breaks a record!

ahh yes. i knew it was some casino downtown. i got the first letter right at least :p

Quote:Wizard154 rolls is the documented record (source). My Ask the Wizard #81 gives the probability of making it to 200 rolls.

It's important to distinguish between a number of rolls without a seven and a number of rolls without a seven-out. It's simple to figure the probability of rolls without a seven; it's just .833333^ n, where n is a number of rolls. I once rolled 56 times without a seven, which has a probability of .0000368, or odds of 27,172 to 1 against. It's also relatively easy to calculate the probability of different numbers of passline decisions without a seven-out, since the probability of a seven-out on any decision is 784/1980, taken from "The Perfect 1980". So, the probability of 60 decisions without a seven-out is (1 - (784/1980) )^ 60 = 7.31 * 10^ -14, or odds of 13,678,568,090,000 to 1 against. Since the average number of rolls in a decision is 3.375, that would come out to roughly 200 rolls (202.5).

However, the Wizard's calculation using "states" is the correct method; it's just a lot more laborious.

Cheers,

Alan Shank

lennyjacky

Quote:goatcabinQuote:Wizard154 rolls is the documented record (source). My Ask the Wizard #81 gives the probability of making it to 200 rolls.

However, the Wizard's calculation using "states" is the correct method; it's just a lot more laborious.

Cheers,

Alan Shank

For those of you that really LOVE working out math problems, there is another formula that can be used that is different from the Wizard's but arrives at the same answers... and it is not recursive or a matrix.

From the article:

"Our aim here is to give an explicit closed-form expression for them, showing that the distribution of L is a linear combination (not a convex combination) of four geometric distributions"

see: http://arxiv.org/abs/0906.1545v2 (for the article)

S. N. Ethier, Fred M. Hoppe

To read the complete article and see the formulas:

you can download a PDF free from the site.

or:http://arxiv.org/pdf/0906.1545v2

(to download just the pdf)

I re-created The Wizard's 200 roll table in excel using their formula in less than 3 minutes.

Quote:tuttigym3. Consecutive Come Out natural winners (7 & 11)?

I don't think this is a record, but I just "rolled" 7 consecutive 7's on a come out roll yesterday (card craps). This was ridiculous as I was betting on the don't pass side the whole time. So I lost 7 in a row. Then I got upset and moved it back to the pass side. Then I 7ed out later on. This is ridiculous.

Quote:focdI don't think this is a record, but I just "rolled" 7 consecutive 7's on a come out roll yesterday (card craps). This was ridiculous as I was betting on the don't pass side the whole time. So I lost 7 in a row. Then I got upset and moved it back to the pass side. Then I 7ed out later on. This is ridiculous.

Odds against seven consecutive 7s are almost 280,000 to 1, if that makes you feel any better. Of course, once you switched, your chances of sevening out on the pass line were not affected. Perhaps it's a good thing that you lost the "dice". >:-)

Where was this unfortunate series of events?

Cheers,

Alan Shank

Woodland, CA

Quote:goatcabinOdds against seven consecutive 7s are almost 280,000 to 1, if that makes you feel any better. Of course, once you switched, your chances of sevening out on the pass line were not affected. Perhaps it's a good thing that you lost the "dice". >:-)

Seriously, I wanted to pass the dice in the middle, but it maybe be perceived rude. It wasn't that I was losing money. When I see patterns I don't like, I just want to quit or pass the dice. It happened in California.

Quote:Wizard154 rolls is the documented record (source). My Ask the Wizard #81 gives the probability of making it to 200 rolls.

Has the "Ask the Wizard" archive been re-indexed? The link to ATW #81 is intact, but the content of the response is different. In this case, the description of the 200 roll probability is missing.

Yes I was counting and had a backup counter aswell. (I counted the total rolls, he counted box numbers hit) I did this in just under 2½ hours (like 2:20 I think).

The table wasn't that full and no one made more than $10k.

Quote:AyecarumbaQuote:Wizard

Has the "Ask the Wizard" archive been re-indexed? The link to ATW #81 is intact, but the content of the response is different. In this case, the description of the 200 roll probability is missing.

Craps introduction mentions the calculation as a in 5.6 billion. (precisely it is 1: 5,590,264,072) Do you want the answer or how to do the calculation? There is a link to the problem in MathProblems.

You can have my spreadsheet using Markov transition matrix if you like. It uses the matrix multiplication function in Excel.

Probability is 1 in 8.81 you will 7-out on 17th roll or higher.

Probability is 1 in 92 you will 7-out on 33rd roll or higher.

Probability is 1 in 10,686 you will 7-out on 65th roll or higher.

I am trying to calculate A^153 where A is a matrix, but Excel does not have that function. So I have to calculate A, A^2, A^4,A^32,A^64,A^128,A^(128+32), etc.

If you use a more sophisticated software (like MatLab or hundreds of other programs) you can calculate it directly. But everyone has EXCEL. You need to read up on matrix multiplication if you didn't learn it, or you forgot it.

Quote:pacomartinQuote:AyecarumbaQuote:Wizard

Has the "Ask the Wizard" archive been re-indexed? The link to ATW #81 is intact, but the content of the response is different. In this case, the description of the 200 roll probability is missing.

Craps introduction mentions the calculation as a in 5.6 billion. (precisely it is 1: 5,590,264,072) Do you want the answer or how to do the calculation? There is a link to the problem in MathProblems.

You can have my spreadsheet using Markov transition matrix if you like. It uses the matrix multiplication function in Excel.

Probability is 1 in 8.81 you will 7-out on 17th roll or higher.

Probability is 1 in 92 you will 7-out on 33rd roll or higher.

Probability is 1 in 10,686 you will 7-out on 65th roll or higher.

I am trying to calculate A^153 where A is a matrix, but Excel does not have that function. So I have to calculate A, A^2, A^4,A^32,A^64,A^128,A^(128+32), etc.

If you use a more sophisticated software (like MatLab or hundreds of other programs) you can calculate it directly. But everyone has EXCEL. You need to read up on matrix multiplication if you didn't learn it, or you forgot it.

Thanks Paco. My original inquiry was the odds of rolling 117 times without throwing a seven (which is apparently the "Sharpshooter Roll-A-Thon" record at the Fremont). I'll check it out on Excel.

Edit: my search came up with the link to Ask the Wizard, but it does not display the craps question (if it is there). Could just be my browser.

Quote:AyecarumbaThanks Paco. My original inquiry was the odds of rolling 117 times without throwing a seven (which is apparently the "Sharpshooter Roll-A-Thon" record at the Fremont). I'll check it out on Excel.

The question of not getting a 7 is a very simple calculation. No matrices are required. You need a matrix multiplication or a equivalently a set of recursive formulas to do the harder problem where you are permitted to roll a 7 on "come-out" rolls.

Probability of rolling n time without getting a 7 is P=(36/30)^n

n=117 1:1,837,408,781

n=123 1:5,486,473,222

Probability of rolling a 7 out on roll 154 or higher 1:5,590,264,072

Quote:AyecarumbaHas the "Ask the Wizard" archive been re-indexed? The link to ATW #81 is intact, but the content of the response is different. In this case, the description of the 200 roll probability is missing.

When the site was revamped in December not everything went perfectly. I'll try to find the missing question.

Looks like it even disappeared from the Craps Q&A page.Quote:riverbedI am eagerly awaiting a reference to that chart.

That has to be a popular page to misplace.

I have a copy of it on my computer.

Here is a link to it

I will remove it once the Wizard gets the info back online.

No Charge :)

The question and table (Probability States in Craps — Recursive Table 1 to 200) is about half way down

"According to the Las Vegas Advisor the record for the longest time a single shooter held the dice in craps is held by Stanley Fujitake, who once held the dice for three hours and six minutes at a downtown casino before sevening out.

(1) What are the odds that Mr. Fujitake could have accomplished his feat, assuming typical craps speed?

(2) What are the odds that this happened in Vegas since 1950?

- Veggie Boy"

Ask the Wizard: Craps

FYI:

riverbed, just go out and roll a 160 roll hand in Las Vegas.

That will get everyone interested in that table again!

added:

Length of a shooters hand

my table using SN Ethier's closed form formula

can be found at his website

Stewart Ethier, Professor

rolls | or more | 1 in | or less | rolls | relative | 1 in |
---|---|---|---|---|---|---|

3 | 0.88888888888888 | 1.1 | 11.1111111111% | 2 | 0.111111111 | 9.0 |

4 | 0.77211934156378 | 1.3 | 22.7880658436% | 3 | 0.116769547 | 8.6 |

5 | 0.66735253772290 | 1.5 | 33.2647462277% | 4 | 0.104766804 | 9.5 |

6 | 0.57612890882995 | 1.7 | 42.3871091170% | 5 | 0.091223629 | 11.0 |

7 | 0.49721087042117 | 2.0 | 50.2789129579% | 6 | 0.078918038 | 12.7 |

8 | 0.42904410662521 | 2.3 | 57.0955893375% | 7 | 0.068166764 | 14.7 |

9 | 0.37019134854117 | 2.7 | 62.9808651459% | 8 | 0.058852758 | 17.0 |

10 | 0.31939069865160 | 3.1 | 68.0609301348% | 9 | 0.05080065 | 19.7 |

11 | 0.27554656198729 | 3.6 | 72.4453438013% | 10 | 0.043844137 | 22.8 |

12 | 0.23771042596129 | 4.2 | 76.2289574039% | 11 | 0.037836136 | 26.4 |

13 | 0.20506192529306 | 4.9 | 79.4938074707% | 12 | 0.032648501 | 30.6 |

14 | 0.17689190346085 | 5.7 | 82.3108096539% | 13 | 0.028170022 | 35.5 |

15 | 0.15258756883984 | 6.6 | 84.7412431160% | 14 | 0.024304335 | 41.1 |

16 | 0.13161956034846 | 7.6 | 86.8380439652% | 15 | 0.020968008 | 47.7 |

17 | 0.11353070335143 | 8.8 | 88.6469296649% | 16 | 0.018088857 | 55.3 |

18 | 0.09792624896423 | 10.2 | 90.2073751036% | 17 | 0.015604454 | 64.1 |

19 | 0.08446541022661 | 11.8 | 91.5534589773% | 18 | 0.013460839 | 74.3 |

20 | 0.07285402926557 | 13.7 | 92.7145970734% | 19 | 0.011611381 | 86.1 |

21 | 0.06283822892249 | 15.9 | 93.7161771078% | 20 | 0.0100158 | 99.8 |

22 | 0.05419891999510 | 18.5 | 94.5801080005% | 21 | 0.008639309 | 115.7 |

23 | 0.04674705117452 | 21.4 | 95.3252948825% | 22 | 0.007451869 | 134.2 |

24 | 0.04031950299040 | 24.8 | 95.9680497010% | 23 | 0.006427548 | 155.6 |

25 | 0.03477553970682 | 28.8 | 96.5224460293% | 24 | 0.005543963 | 180.4 |

rolls | or more | 1 in | or less | rolls | relative | 1 in |

26 | 0.02999374426323 | 33.3 | 97.0006255737% | 25 | 0.004781795 | 209.1 |

27 | 0.02586937115903 | 38.7 | 97.4130628841% | 26 | 0.004124373 | 242.5 |

28 | 0.02231206077227 | 44.8 | 97.7687939228% | 27 | 0.00355731 | 281.1 |

29 | 0.01924386611177 | 52.0 | 98.0756133888% | 28 | 0.003068195 | 325.9 |

30 | 0.01659754954925 | 60.2 | 98.3402450451% | 29 | 0.002646317 | 377.9 |

31 | 0.01431511277769 | 69.9 | 98.5684887222% | 30 | 0.002282437 | 438.1 |

32 | 0.01234652819497 | 81.0 | 98.7653471805% | 31 | 0.001968585 | 508.0 |

33 | 0.01064864421152 | 93.9 | 98.9351355788% | 32 | 0.001697884 | 589.0 |

34 | 0.00918424070887 | 108.9 | 99.0815759291% | 33 | 0.001464404 | 682.9 |

35 | 0.00792121410618 | 126.2 | 99.2078785894% | 34 | 0.001263027 | 791.7 |

36 | 0.00683187428823 | 146.4 | 99.3168125712% | 35 | 0.00108934 | 918.0 |

37 | 0.00589233806802 | 169.7 | 99.4107661932% | 36 | 0.000939536 | 1,064.4 |

38 | 0.00508200594980 | 196.8 | 99.4917994050% | 37 | 0.000810332 | 1,234.1 |

39 | 0.00438311076688 | 228.1 | 99.5616889233% | 38 | 0.000698895 | 1,430.8 |

40 | 0.00378032833215 | 264.5 | 99.6219671668% | 39 | 0.000602782 | 1,659.0 |

41 | 0.00326044158919 | 306.7 | 99.6739558411% | 40 | 0.000519887 | 1,923.5 |

42 | 0.00281205091828 | 355.6 | 99.7187949082% | 41 | 0.000448391 | 2,230.2 |

43 | 0.00242532425864 | 412.3 | 99.7574675741% | 42 | 0.000386727 | 2,585.8 |

44 | 0.00209178157754 | 478.1 | 99.7908218422% | 43 | 0.000333543 | 2,998.1 |

45 | 0.00180410896744 | 554.3 | 99.8195891033% | 44 | 0.000287673 | 3,476.2 |

46 | 0.00155599830003 | 642.7 | 99.8444001700% | 45 | 0.000248111 | 4,030.5 |

47 | 0.00134200892494 | 745.2 | 99.8657991075% | 46 | 0.000213989 | 4,673.1 |

48 | 0.00115744838330 | 864.0 | 99.8842551617% | 47 | 0.000184561 | 5,418.3 |

49 | 0.00099826952247 | 1,001.7 | 99.9001730478% | 48 | 0.000159179 | 6,282.2 |

50 | 0.00086098175742 | 1,161.5 | 99.9139018243% | 49 | 0.000137288 | 7,284.0 |

rolls | or more | 1 in | or less | rolls | relative | 1 in |

51 | 0.00074257453382 | 1,346.7 | 99.9257425466% | 50 | 0.000118407 | 8,445.4 |

52 | 0.00064045131559 | 1,561.4 | 99.9359548684% | 51 | 0.000102123 | 9,792.1 |

53 | 0.00055237264960 | 1,810.4 | 99.9447627350% | 52 | 8.80787E-05 | 11,353.5 |

54 | 0.00047640705978 | 2,099.0 | 99.9523592940% | 53 | 7.59656E-05 | 13,163.9 |

55 | 0.00041088869386 | 2,433.7 | 99.9589111306% | 54 | 6.55184E-05 | 15,262.9 |

56 | 0.00035438079441 | 2,821.8 | 99.9645619206% | 55 | 5.65079E-05 | 17,696.6 |

57 | 0.00030564419326 | 3,271.8 | 99.9694355807% | 56 | 4.87366E-05 | 20,518.5 |

58 | 0.00026361013840 | 3,793.5 | 99.9736389862% | 57 | 4.20341E-05 | 23,790.2 |

59 | 0.00022735685786 | 4,398.4 | 99.9772643142% | 58 | 3.62533E-05 | 27,583.7 |

60 | 0.00019608934644 | 5,099.7 | 99.9803910654% | 59 | 3.12675E-05 | 31,982.1 |

61 | 0.00016912193225 | 5,912.9 | 99.9830878068% | 60 | 2.69674E-05 | 37,081.8 |

62 | 0.00014586324068 | 6,855.7 | 99.9854136759% | 61 | 2.32587E-05 | 42,994.7 |

63 | 0.00012580322626 | 7,948.9 | 99.9874196774% | 62 | 2.006E-05 | 49,850.4 |

64 | 0.00010850198780 | 9,216.4 | 99.9891498012% | 63 | 1.73012E-05 | 57,799.3 |

65 | 0.00009358012187 | 10,686.0 | 99.9906419878% | 64 | 1.49219E-05 | 67,015.7 |

66 | 0.00008071040278 | 12,390.0 | 99.9919289597% | 65 | 1.28697E-05 | 77,701.8 |

67 | 0.00006961060683 | 14,365.6 | 99.9930389393% | 66 | 1.10998E-05 | 90,091.7 |

68 | 0.00006003732342 | 16,656.3 | 99.9939962677% | 67 | 9.57328E-06 | 104,457.4 |

69 | 0.00005178061719 | 19,312.2 | 99.9948219383% | 68 | 8.25671E-06 | 121,113.7 |

70 | 0.00004465942437 | 22,391.7 | 99.9955340576% | 69 | 7.12119E-06 | 140,425.9 |

71 | 0.00003851758217 | 25,962.2 | 99.9961482418% | 70 | 6.14184E-06 | 162,817.6 |

72 | 0.00003322040425 | 30,102.0 | 99.9966779596% | 71 | 5.29718E-06 | 188,779.8 |

73 | 0.00002865172718 | 34,901.9 | 99.9971348273% | 72 | 4.56868E-06 | 218,881.7 |

74 | 0.00002471136299 | 40,467.2 | 99.9975288637% | 73 | 3.94036E-06 | 253,783.6 |

75 | 0.00002131290221 | 46,919.9 | 99.9978687098% | 74 | 3.39846E-06 | 294,250.9 |

rolls | or more | 1 in | or less | rolls | relative | 1 in |

76 | 0.00001838181893 | 54,401.6 | 99.9981618181% | 75 | 2.93108E-06 | 341,170.8 |

77 | 0.00001585383648 | 63,076.2 | 99.9984146164% | 76 | 2.52798E-06 | 395,572.4 |

78 | 0.00001367351794 | 73,134.1 | 99.9986326482% | 77 | 2.18032E-06 | 458,648.6 |

79 | 0.00001179305039 | 84,795.7 | 99.9988206950% | 78 | 1.88047E-06 | 531,782.6 |

80 | 0.00001017119648 | 98,316.9 | 99.9989828804% | 79 | 1.62185E-06 | 616,578.3 |

81 | 0.00000877239003 | 113,994.0 | 99.9991227610% | 80 | 1.39881E-06 | 714,895.2 |

82 | 0.00000756595617 | 132,171.0 | 99.9992434044% | 81 | 1.20643E-06 | 828,889.2 |

83 | 0.00000652543863 | 153,246.4 | 99.9993474561% | 82 | 1.04052E-06 | 961,060.2 |

84 | 0.00000562801955 | 177,682.4 | 99.9994371980% | 83 | 8.97419E-07 | 1,114,306.6 |

85 | 0.00000485401915 | 206,014.8 | 99.9995145981% | 84 | 7.74E-07 | 1,291,989.0 |

86 | 0.00000418646412 | 238,865.1 | 99.9995813536% | 85 | 6.67555E-07 | 1,498,003.8 |

87 | 0.00000361071542 | 276,953.4 | 99.9996389285% | 86 | 5.75749E-07 | 1,736,868.9 |

88 | 0.00000311414728 | 321,115.2 | 99.9996885853% | 87 | 4.96568E-07 | 2,013,822.3 |

89 | 0.00000268587029 | 372,318.8 | 99.9997314130% | 88 | 4.28277E-07 | 2,334,937.5 |

90 | 0.00000231649263 | 431,687.1 | 99.9997683507% | 89 | 3.69378E-07 | 2,707,256.3 |

91 | 0.00000199791409 | 500,522.0 | 99.9998002086% | 90 | 3.18579E-07 | 3,138,943.4 |

92 | 0.00000172314846 | 580,333.0 | 99.9998276852% | 91 | 2.74766E-07 | 3,639,465.4 |

93 | 0.00000148617032 | 672,870.4 | 99.9998513830% | 92 | 2.36978E-07 | 4,219,798.5 |

94 | 0.00000128178289 | 780,163.3 | 99.9998718217% | 93 | 2.04387E-07 | 4,892,668.8 |

95 | 0.00000110550410 | 904,564.7 | 99.9998894496% | 94 | 1.76279E-07 | 5,672,832.2 |

96 | 0.00000095346827 | 1,048,802.6 | 99.9999046532% | 95 | 1.52036E-07 | 6,577,396.9 |

97 | 0.00000082234136 | 1,216,040.0 | 99.9999177659% | 96 | 1.31127E-07 | 7,626,199.5 |

98 | 0.00000070924784 | 1,409,944.4 | 99.9999290752% | 97 | 1.13094E-07 | 8,842,239.5 |

99 | 0.00000061170764 | 1,634,767.9 | 99.9999388292% | 98 | 9.75402E-08 | 10,252,183.9 |

100 | 0.00000052758178 | 1,895,440.7 | 99.9999472418% | 99 | 8.41259E-08 | 11,886,951.7 |

rolls | or more | 1 in | or less | rolls | relative | 1 in |

101 | 0.00000045502544 | 2,197,679.3 | 99.9999544975% | 100 | 7.25563E-08 | 13,782,392.5 |

102 | 0.00000039244750 | 2,548,111.5 | 99.9999607553% | 101 | 6.25779E-08 | 15,980,071.7 |

103 | 0.00000033847567 | 2,954,422.1 | 99.9999661524% | 102 | 5.39718E-08 | 18,528,183.3 |

104 | 0.00000029192638 | 3,425,521.2 | 99.9999708074% | 103 | 4.65493E-08 | 21,482,605.4 |

105 | 0.00000025177884 | 3,971,739.7 | 99.9999748221% | 104 | 4.01475E-08 | 24,908,126.6 |

106 | 0.00000021715264 | 4,605,055.7 | 99.9999782847% | 105 | 3.46262E-08 | 28,879,866.3 |

107 | 0.00000018728845 | 5,339,357.5 | 99.9999812712% | 106 | 2.98642E-08 | 33,484,921.9 |

108 | 0.00000016153137 | 6,190,747.8 | 99.9999838469% | 107 | 2.57571E-08 | 38,824,279.5 |

109 | 0.00000013931657 | 7,177,897.1 | 99.9999860683% | 108 | 2.22148E-08 | 45,015,027.1 |

110 | 0.00000012015688 | 8,322,452.8 | 99.9999879843% | 109 | 1.91597E-08 | 52,192,924.4 |

111 | 0.00000010363216 | 9,649,514.3 | 99.9999896368% | 110 | 1.65247E-08 | 60,515,377.1 |

112 | 0.00000008938002 | 11,188,183.1 | 99.9999910620% | 111 | 1.42521E-08 | 70,164,891.6 |

113 | 0.00000007708792 | 12,972,201.3 | 99.9999922912% | 112 | 1.22921E-08 | 81,353,074.4 |

114 | 0.00000006648631 | 15,040,691.1 | 99.9999933514% | 113 | 1.06016E-08 | 94,325,275.8 |

115 | 0.00000005734269 | 17,439,013.2 | 99.9999942657% | 114 | 9.14361E-09 | 109,365,966.7 |

116 | 0.00000004945657 | 20,219,761.1 | 99.9999950543% | 115 | 7.88613E-09 | 126,804,979.5 |

117 | 0.00000004265499 | 23,443,914.7 | 99.9999957345% | 116 | 6.80158E-09 | 147,024,741.8 |

118 | 0.00000003678881 | 27,182,177.6 | 99.9999963211% | 117 | 5.86618E-09 | 170,468,654.7 |

119 | 0.00000003172938 | 31,516,527.4 | 99.9999968271% | 118 | 5.05943E-09 | 197,650,832.5 |

120 | 0.00000002736576 | 36,542,013.4 | 99.9999972634% | 119 | 4.36362E-09 | 229,167,361.7 |

121 | 0.00000002360225 | 42,368,841.1 | 99.9999976398% | 120 | 3.76351E-09 | 265,709,376.6 |

122 | 0.00000002035632 | 49,124,789.1 | 99.9999979644% | 121 | 3.24593E-09 | 308,078,214.0 |

123 | 0.00000001755679 | 56,958,010.6 | 99.9999982443% | 122 | 2.79953E-09 | 357,203,007.8 |

124 | 0.00000001514227 | 66,040,282.9 | 99.9999984858% | 123 | 2.41452E-09 | 414,161,006.8 |

125 | 0.00000001305981 | 76,570,774.2 | 99.9999986940% | 124 | 2.08246E-09 | 480,201,304.5 |

rolls | or more | 1 in | or less | rolls | relative | 1 in |

126 | 0.00000001126375 | 88,780,411.1 | 99.9999988736% | 125 | 1.79607E-09 | 556,772,080.0 |

127 | 0.00000000971469 | 102,936,942.6 | 99.9999990285% | 126 | 1.54906E-09 | 645,552,466.6 |

128 | 0.00000000837866 | 119,350,812.1 | 99.9999991621% | 127 | 1.33602E-09 | 748,489,447.2 |

129 | 0.00000000722638 | 138,381,964.5 | 99.9999992774% | 128 | 1.15229E-09 | 867,840,236.0 |

130 | 0.00000000623256 | 160,447,740.2 | 99.9999993767% | 129 | 9.93816E-10 | 1,006,222,112.8 |

131 | 0.00000000537542 | 186,032,027.0 | 99.9999994625% | 130 | 8.5714E-10 | 1,166,670,067.3 |

132 | 0.00000000463616 | 215,695,870.9 | 99.9999995364% | 131 | 7.39261E-10 | 1,352,701,845.4 |

133 | 0.00000000399856 | 250,089,780.1 | 99.9999996001% | 132 | 6.37593E-10 | 1,568,397,883.1 |

134 | 0.00000000344866 | 289,967,989.9 | 99.9999996551% | 133 | 5.49908E-10 | 1,818,487,443.3 |

135 | 0.00000000297438 | 336,205,002.5 | 99.9999997026% | 134 | 4.74281E-10 | 2,108,455,911.4 |

136 | 0.00000000256532 | 389,814,764.7 | 99.9999997435% | 135 | 4.09055E-10 | 2,444,660,286.0 |

137 | 0.00000000221252 | 451,972,902.3 | 99.9999997787% | 136 | 3.52799E-10 | 2,834,475,948.2 |

138 | 0.00000000190824 | 524,042,501.5 | 99.9999998092% | 137 | 3.0428E-10 | 3,286,447,400.4 |

139 | 0.00000000164581 | 607,604,000.1 | 99.9999998354% | 138 | 2.62433E-10 | 3,810,492,076.4 |

140 | 0.00000000141947 | 704,489,845.5 | 99.9999998581% | 139 | 2.26342E-10 | 4,418,094,044.3 |

141 | 0.00000000122425 | 816,824,679.2 | 99.9999998776% | 140 | 1.95214E-10 | 5,122,584,573.0 |

142 | 0.00000000105589 | 947,071,928.4 | 99.9999998944% | 141 | 1.68367E-10 | 5,939,410,552.6 |

143 | 0.00000000091067 | 1,098,087,827.7 | 99.9999999089% | 142 | 1.45212E-10 | 6,886,480,147.4 |

144 | 0.00000000078543 | 1,273,184,054.1 | 99.9999999215% | 143 | 1.25242E-10 | 7,984,567,755.0 |

145 | 0.00000000067741 | 1,476,200,350.1 | 99.9999999323% | 144 | 1.08018E-10 | 9,257,751,028.6 |

146 | 0.00000000058425 | 1,711,588,726.3 | 99.9999999416% | 145 | 9.31623E-11 | 10,733,960,793.7 |

147 | 0.00000000050390 | 1,984,511,091.5 | 99.9999999496% | 146 | 8.03501E-11 | 12,445,541,431.6 |

148 | 0.00000000043460 | 2,300,952,449.5 | 99.9999999565% | 147 | 6.92998E-11 | 14,430,058,546.8 |

149 | 0.00000000037483 | 2,667,852,146.3 | 99.9999999625% | 148 | 5.97693E-11 | 16,730,997,178.0 |

150 | 0.00000000032328 | 3,093,256,045.4 | 99.9999999677% | 149 | 5.15494E-11 | 19,398,856,069.4 |

rolls | or more | 1 in | or less | rolls | relative | 1 in |

151 | 0.00000000027882 | 3,586,492,968.0 | 99.9999999721% | 150 | 4.446E-11 | 22,492,132,184.8 |

152 | 0.00000000024048 | 4,158,379,267.9 | 99.9999999760% | 151 | 3.83457E-11 | 26,078,570,573.7 |

153 | 0.00000000020741 | 4,821,456,026.8 | 99.9999999793% | 152 | 3.30721E-11 | 30,236,966,550.2 |

154 | 0.00000000017888 | 5,590,264,071.8 | 99.9999999821% | 153 | 2.85237E-11 | 35,058,517,488.9 |

155 | 0.00000000015428 | 6,481,662,846.1 | 99.9999999846% | 154 | 2.46011E-11 | 40,648,590,642.7 |

156 | 0.00000000013306 | 7,515,200,124.7 | 99.9999999867% | 155 | 2.12177E-11 | 47,130,474,563.3 |

157 | 0.00000000011476 | 8,713,540,684.9 | 99.9999999885% | 156 | 1.82998E-11 | 54,645,387,700.9 |

158 | 0.00000000009898 | 10,102,963,328.6 | 99.9999999901% | 157 | 1.5783E-11 | 63,359,143,891.4 |

159 | 0.00000000008537 | 11,713,937,159.4 | 99.9999999915% | 158 | 1.36124E-11 | 73,462,191,132.4 |

160 | 0.00000000007363 | 13,581,789,749.4 | 99.9999999926% | 159 | 1.17404E-11 | 85,176,071,932.7 |

161 | 0.00000000006350 | 15,747,481,848.9 | 99.9999999936% | 160 | 1.01258E-11 | 98,757,735,373.5 |

162 | 0.00000000005477 | 18,258,505,628.3 | 99.9999999945% | 161 | 8.73324E-12 | 114,505,088,285.8 |

163 | 0.00000000004724 | 21,169,926,149.2 | 99.9999999953% | 162 | 7.5322E-12 | 132,763,387,399.6 |

164 | 0.00000000004074 | 24,545,588,904.6 | 99.9999999959% | 163 | 6.49625E-12 | 153,935,010,249.7 |

165 | 0.00000000003514 | 28,459,519,907.0 | 99.9999999965% | 164 | 5.60285E-12 | 178,480,546,402.4 |

166 | 0.00000000003031 | 32,997,549,029.4 | 99.9999999970% | 165 | 4.83236E-12 | 206,938,364,534.8 |

167 | 0.00000000002614 | 38,259,192,196.8 | 99.9999999974% | 166 | 4.16778E-12 | 239,936,048,341.5 |

168 | 0.00000000002254 | 44,359,833,703.0 | 99.9999999977% | 167 | 3.59457E-12 | 278,197,462,851.4 |

169 | 0.00000000001944 | 51,433,256,510.9 | 99.9999999981% | 168 | 3.1003E-12 | 322,549,659,972.8 |

170 | 0.00000000001677 | 59,634,576,022.7 | 99.9999999983% | 169 | 2.67386E-12 | 373,991,000,446.0 |

171 | 0.00000000001446 | 69,143,641,656.3 | 99.9999999986% | 170 | 2.30616E-12 | 433,622,147,830.8 |

172 | 0.00000000001247 | 80,168,980,822.0 | 99.9999999988% | 171 | 1.98896E-12 | 502,774,169,954.8 |

173 | 0.00000000001076 | 92,952,371,788.4 | 99.9999999989% | 172 | 1.71552E-12 | 582,914,784,800.7 |

174 | 0.00000000000928 | 107,774,145,717.0 | 99.9999999991% | 173 | 1.47948E-12 | 675,911,695,538.1 |

175 | 0.00000000000800 | 124,959,334,135.9 | 99.9999999992% | 174 | 1.27609E-12 | 783,643,575,321.1 |

rolls | or more | 1 in | or less | rolls | relative | 1 in |

176 | 0.00000000000690 | 144,884,796,662.5 | 99.9999999993% | 175 | 1.10056E-12 | 908,624,962,649.1 |

177 | 0.00000000000595 | 167,987,485,281.4 | 99.9999999994% | 176 | 9.49241E-13 | 1,053,473,597,045.7 |

178 | 0.00000000000513 | 194,774,026,407.4 | 99.9999999995% | 177 | 8.18678E-13 | 1,221,480,777,697.5 |

179 | 0.00000000000443 | 225,831,830,861.8 | 99.9999999996% | 178 | 7.05991E-13 | 1,416,449,010,023.7 |

180 | 0.00000000000382 | 261,841,975,396.1 | 99.9999999996% | 179 | 6.09068E-13 | 1,641,851,851,028.3 |

181 | 0.00000000000329 | 303,594,138,247.5 | 99.9999999997% | 180 | 5.25135E-13 | 1,904,270,455,547.8 |

182 | 0.00000000000284 | 352,003,916,250.7 | 99.9999999997% | 181 | 4.53082E-13 | 2,207,105,918,829.0 |

183 | 0.00000000000245 | 408,132,903,260.5 | 99.9999999998% | 182 | 3.90687E-13 | 2,559,590,581,057.4 |

184 | 0.00000000000211 | 473,211,970,190.7 | 99.9999999998% | 183 | 3.36953E-13 | 2,967,775,701,726.9 |

185 | 0.00000000000182 | 548,668,257,185.0 | 99.9999999998% | 184 | 2.90656E-13 | 3,440,488,638,174.6 |

186 | 0.00000000000157 | 636,156,469,839.7 | 99.9999999998% | 185 | 2.50577E-13 | 3,990,783,896,650.9 |

187 | 0.00000000000136 | 737,595,165,784.2 | 99.9999999999% | 186 | 2.1616E-13 | 4,626,193,762,065.2 |

188 | 0.00000000000117 | 855,208,827,358.7 | 99.9999999999% | 187 | 1.86517E-13 | 5,361,428,127,822.0 |

189 | 0.00000000000101 | 991,576,643,014.7 | 99.9999999999% | 188 | 1.6076E-13 | 6,220,441,474,268.6 |

190 | 0.00000000000087 | 1,149,689,067,182.7 | 99.9999999999% | 189 | 1.38778E-13 | 7,205,759,403,792.8 |

191 | 0.00000000000075 | 1,333,013,398,924.6 | 99.9999999999% | 190 | 1.19571E-13 | 8,363,230,505,794.8 |

192 | 0.00000000000065 | 1,545,569,817,469.8 | 99.9999999999% | 191 | 1.0314E-13 | 9,695,585,850,098.0 |

193 | 0.00000000000056 | 1,792,019,542,039.8 | 99.9999999999% | 192 | 8.90399E-14 | 11,230,921,764,016.2 |

194 | 0.00000000000048 | 2,077,767,049,248.8 | 100.0000000000% | 193 | 7.67164E-14 | 13,035,020,629,147.6 |

195 | 0.00000000000042 | 2,409,078,589,639.8 | 100.0000000000% | 194 | 6.61693E-14 | 15,112,750,427,417.8 |

196 | 0.00000000000036 | 2,793,219,602,341.5 | 100.0000000000% | 195 | 5.70655E-14 | 17,523,733,958,640.1 |

197 | 0.00000000000031 | 3,238,614,041,259.3 | 100.0000000000% | 196 | 4.92939E-14 | 20,286,484,807,975.2 |

198 | 0.00000000000027 | 3,755,029,106,716.0 | 100.0000000000% | 197 | 4.24105E-14 | 23,579,055,640,683.2 |

199 | 0.00000000000023 | 4,353,789,433,581.8 | 100.0000000000% | 198 | 3.66374E-14 | 27,294,543,196,184.8 |

200 | 0.00000000000020 | 5,048,025,432,896.4 | 100.0000000000% | 199 | 3.16414E-14 | 31,604,207,911,371.9 |

201 | 0.00000000000017 | 5,852,961,232,947.0 | 100.0000000000% | 200 | 2.72005E-14 | 36,764,078,590,779.6 |

Just the other day a lady on my right rolled 63 times, which is only one in 7949 shooters, fewer than I thought. The probability really jumps along as the roll lengthens. I used to chart the intervals of the sevens, and I found that process quite instructive in understanding the game. I don't think that very many players that I have played with realize that half the shooters roll 8.5 times or less, and this has profound implications for your betting and money management strategies. I notice from the chart that the proportion of any roll number to the previous one is .8562 or so, whereas the intervals between sevens were at a proportion to the next interval of .83333, a 6:5 ratio. I guess the difference is the seven-out/come-out factor.

Another question: I notice that the median roll in 8.5, yet the probability of 2 in the table (I assume this might be expressed as 'one chance in two') is right at 7. Why the difference?

Nice observations.Quote:riverbedThank you so much! I've been looking for this chart for a couple of weeks. And I certainly do intend to go out and roll a 160 roll hand as soon as possible.

Just the other day a lady on my right rolled 63 times, which is only one in 7949 shooters, fewer than I thought. The probability really jumps along as the roll lengthens. I used to chart the intervals of the sevens, and I found that process quite instructive in understanding the game. I don't think that very many players that I have played with realize that half the shooters roll 8.5 times or less, and this has profound implications for your betting and money management strategies. I notice from the chart that the proportion of any roll number to the previous one is .8562 or so, whereas the intervals between sevens were at a proportion to the next interval of .83333, a 6:5 ratio. I guess the difference is the seven-out/come-out factor.

The ratio for the 7 is always the same.

For the shooter 7out, there is a difference in the first few iterations but as you have shown once past that, the ratio converges very quickly.

A transition matrix also shows this.

No, the median is not 8.5 rollsQuote:riverbedAnother question: I notice that the median roll in 8.5, yet the probability of 2 in the table (I assume this might be expressed as 'one chance in two') is right at 7. Why the difference?

8.5 is the average number of rolls per shooter (8.525510204 or 1671/196 to be exact)

6 is the median. (look at my above table or the photo below)

about ~50% (50.2789129579%) of all shooters will 7out by the 6th roll so the other half will get past the 6th roll.

length of a shooter's hand - cumulative probability

The matrix algebra program does not have to be sophisticated.Quote:pacomartinYou can have my spreadsheet using Markov transition matrix if you like. It uses the matrix multiplication function in Excel.

I am trying to calculate A^153 where A is a matrix, but Excel does not have that function. So I have to calculate A, A^2, A^4,A^32,A^64,A^128,A^(128+32), etc.

If you use a more sophisticated software (like MatLab or hundreds of other programs) you can calculate it directly. But everyone has EXCEL. You need to read up on matrix multiplication if you didn't learn it, or you forgot it.

One needs to use something else than Excel sledgehammer methods.

I am not a Matrix Algebra expert by any means, some may think I am, unlike ChrystalMath

A good lesson here

Markov Systems

FREE online program

JavaScript. I have yet to see this one fail.

Matrix Algebra Tool

I thought I posted this here before, maybe at another forum.Quote:pacomartin

Craps introduction mentions the calculation as a in 5.6 billion. (precisely it is 1: 5,590,264,072) Do you want the answer or how to do the calculation? There is a link to the problem in MathProblems.

You can have my spreadsheet using Markov transition matrix if you like. It uses the matrix multiplication function in Excel.

I am trying to calculate A^153 where A is a matrix, but Excel does not have that function. So I have to calculate A, A^2, A^4,A^32,A^64,A^128,A^(128+32), etc.

If you use a more sophisticated software (like MatLab or hundreds of other programs) you can calculate it directly. But everyone has EXCEL. You need to read up on matrix multiplication if you didn't learn it, or you forgot it.

One can figure this in Excel with a simple closed form formula (no matrix multiplication needed)

Quite easy actually.

closed form formula using eigenvectors and eigenvalues (remember math class with matrix algebra???)

The explicit closed-form expression for the length of a craps shooter hand is:

=(_c1*(e1_^(Roll))+(_c2*(e2_^(Roll))+(_c3*(e3_)^(Roll))+(_c4*(e4_)^(Roll))))

values

c1 1.211844812464510000

c2 -0.006375542263784770

c3 -0.004042671248651500

c4 -0.201426598952082000

e1 0.862473751659322000

e2 0.741708271459795000

e3 0.709206775794379000

e4 0.186611201086502000

From the pdf:

A world record in Atlantic City

and the length of the shooter’s hand at craps

S. N. Ethier and Fred M. Hoppe

length of the shooter’s hand at craps free pdf

yes, this is sick math.

These guys had nothing better to do one afternoon!

Enjoy!

silly

Sally

Again, many thanks.

The distribution. The probabilities of each possible outcome.Quote:riverbedMany thanks. But what accounts for the median and the average rolls per shooter being different?

It is not normal where the mean (average), median and mode are about the same.

here is a photo of the relative probabilities

One can see the mode is equal to 3.

We can not see the median unless we look at the cumulative graph.

And the average is nowhere near a peak.

In most geometric type distributions, the median is less than the mean.

crank out the old stats book for more

On average there is a comeout roll every 5.06 rolls, so 4.06/5.06 = 0.802 rolls aren’t comeout rolls. For each of these rolls there’s a 5 in 6, or 1 in 1.2, chance a seven won’t be rolled.

3.131 * 1.159456^(n-10)

3.131 being the value for 10, 1.159456 being the sustained rate for each iteration after 10.

So the value for 154 is 3.131 * 1.159456^144 = 5.601 x 10^9