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riverbed
riverbed
Joined: Jan 14, 2010
  • Threads: 4
  • Posts: 21
June 13th, 2012 at 7:19:48 PM permalink
I am eagerly awaiting a reference to that chart.
7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
Thanks for this post from:
ChumpChange
June 13th, 2012 at 8:15:10 PM permalink
Quote: riverbed

I am eagerly awaiting a reference to that chart.

Looks like it even disappeared from the Craps Q&A page.
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

rollsor more1 inor lessrollsrelative1 in
30.888888888888881.111.1111111111%20.1111111119.0
40.772119341563781.322.7880658436%30.1167695478.6
50.667352537722901.533.2647462277%40.1047668049.5
60.576128908829951.742.3871091170%50.09122362911.0
70.497210870421172.050.2789129579%60.07891803812.7
80.429044106625212.357.0955893375%70.06816676414.7
90.370191348541172.762.9808651459%80.05885275817.0
100.319390698651603.168.0609301348%90.0508006519.7
110.275546561987293.672.4453438013%100.04384413722.8
120.237710425961294.276.2289574039%110.03783613626.4
130.205061925293064.979.4938074707%120.03264850130.6
140.176891903460855.782.3108096539%130.02817002235.5
150.152587568839846.684.7412431160%140.02430433541.1
160.131619560348467.686.8380439652%150.02096800847.7
170.113530703351438.888.6469296649%160.01808885755.3
180.0979262489642310.290.2073751036%170.01560445464.1
190.0844654102266111.891.5534589773%180.01346083974.3
200.0728540292655713.792.7145970734%190.01161138186.1
210.0628382289224915.993.7161771078%200.010015899.8
220.0541989199951018.594.5801080005%210.008639309115.7
230.0467470511745221.495.3252948825%220.007451869134.2
240.0403195029904024.895.9680497010%230.006427548155.6
250.0347755397068228.896.5224460293%240.005543963180.4
rollsor more1 inor lessrollsrelative1 in
260.0299937442632333.397.0006255737%250.004781795209.1
270.0258693711590338.797.4130628841%260.004124373242.5
280.0223120607722744.897.7687939228%270.00355731281.1
290.0192438661117752.098.0756133888%280.003068195325.9
300.0165975495492560.298.3402450451%290.002646317377.9
310.0143151127776969.998.5684887222%300.002282437438.1
320.0123465281949781.098.7653471805%310.001968585508.0
330.0106486442115293.998.9351355788%320.001697884589.0
340.00918424070887108.999.0815759291%330.001464404682.9
350.00792121410618126.299.2078785894%340.001263027791.7
360.00683187428823146.499.3168125712%350.00108934918.0
370.00589233806802169.799.4107661932%360.0009395361,064.4
380.00508200594980196.899.4917994050%370.0008103321,234.1
390.00438311076688228.199.5616889233%380.0006988951,430.8
400.00378032833215264.599.6219671668%390.0006027821,659.0
410.00326044158919306.799.6739558411%400.0005198871,923.5
420.00281205091828355.699.7187949082%410.0004483912,230.2
430.00242532425864412.399.7574675741%420.0003867272,585.8
440.00209178157754478.199.7908218422%430.0003335432,998.1
450.00180410896744554.399.8195891033%440.0002876733,476.2
460.00155599830003642.799.8444001700%450.0002481114,030.5
470.00134200892494745.299.8657991075%460.0002139894,673.1
480.00115744838330864.099.8842551617%470.0001845615,418.3
490.000998269522471,001.799.9001730478%480.0001591796,282.2
500.000860981757421,161.599.9139018243%490.0001372887,284.0
rollsor more1 inor lessrollsrelative1 in
510.000742574533821,346.799.9257425466%500.0001184078,445.4
520.000640451315591,561.499.9359548684%510.0001021239,792.1
530.000552372649601,810.499.9447627350%528.80787E-0511,353.5
540.000476407059782,099.099.9523592940%537.59656E-0513,163.9
550.000410888693862,433.799.9589111306%546.55184E-0515,262.9
560.000354380794412,821.899.9645619206%555.65079E-0517,696.6
570.000305644193263,271.899.9694355807%564.87366E-0520,518.5
580.000263610138403,793.599.9736389862%574.20341E-0523,790.2
590.000227356857864,398.499.9772643142%583.62533E-0527,583.7
600.000196089346445,099.799.9803910654%593.12675E-0531,982.1
610.000169121932255,912.999.9830878068%602.69674E-0537,081.8
620.000145863240686,855.799.9854136759%612.32587E-0542,994.7
630.000125803226267,948.999.9874196774%622.006E-0549,850.4
640.000108501987809,216.499.9891498012%631.73012E-0557,799.3
650.0000935801218710,686.099.9906419878%641.49219E-0567,015.7
660.0000807104027812,390.099.9919289597%651.28697E-0577,701.8
670.0000696106068314,365.699.9930389393%661.10998E-0590,091.7
680.0000600373234216,656.399.9939962677%679.57328E-06104,457.4
690.0000517806171919,312.299.9948219383%688.25671E-06121,113.7
700.0000446594243722,391.799.9955340576%697.12119E-06140,425.9
710.0000385175821725,962.299.9961482418%706.14184E-06162,817.6
720.0000332204042530,102.099.9966779596%715.29718E-06188,779.8
730.0000286517271834,901.999.9971348273%724.56868E-06218,881.7
740.0000247113629940,467.299.9975288637%733.94036E-06253,783.6
750.0000213129022146,919.999.9978687098%743.39846E-06294,250.9
rollsor more1 inor lessrollsrelative1 in
760.0000183818189354,401.699.9981618181%752.93108E-06341,170.8
770.0000158538364863,076.299.9984146164%762.52798E-06395,572.4
780.0000136735179473,134.199.9986326482%772.18032E-06458,648.6
790.0000117930503984,795.799.9988206950%781.88047E-06531,782.6
800.0000101711964898,316.999.9989828804%791.62185E-06616,578.3
810.00000877239003113,994.099.9991227610%801.39881E-06714,895.2
820.00000756595617132,171.099.9992434044%811.20643E-06828,889.2
830.00000652543863153,246.499.9993474561%821.04052E-06961,060.2
840.00000562801955177,682.499.9994371980%838.97419E-071,114,306.6
850.00000485401915206,014.899.9995145981%847.74E-071,291,989.0
860.00000418646412238,865.199.9995813536%856.67555E-071,498,003.8
870.00000361071542276,953.499.9996389285%865.75749E-071,736,868.9
880.00000311414728321,115.299.9996885853%874.96568E-072,013,822.3
890.00000268587029372,318.899.9997314130%884.28277E-072,334,937.5
900.00000231649263431,687.199.9997683507%893.69378E-072,707,256.3
910.00000199791409500,522.099.9998002086%903.18579E-073,138,943.4
920.00000172314846580,333.099.9998276852%912.74766E-073,639,465.4
930.00000148617032672,870.499.9998513830%922.36978E-074,219,798.5
940.00000128178289780,163.399.9998718217%932.04387E-074,892,668.8
950.00000110550410904,564.799.9998894496%941.76279E-075,672,832.2
960.000000953468271,048,802.699.9999046532%951.52036E-076,577,396.9
970.000000822341361,216,040.099.9999177659%961.31127E-077,626,199.5
980.000000709247841,409,944.499.9999290752%971.13094E-078,842,239.5
990.000000611707641,634,767.999.9999388292%989.75402E-0810,252,183.9
1000.000000527581781,895,440.799.9999472418%998.41259E-0811,886,951.7
rollsor more1 inor lessrollsrelative1 in
1010.000000455025442,197,679.399.9999544975%1007.25563E-0813,782,392.5
1020.000000392447502,548,111.599.9999607553%1016.25779E-0815,980,071.7
1030.000000338475672,954,422.199.9999661524%1025.39718E-0818,528,183.3
1040.000000291926383,425,521.299.9999708074%1034.65493E-0821,482,605.4
1050.000000251778843,971,739.799.9999748221%1044.01475E-0824,908,126.6
1060.000000217152644,605,055.799.9999782847%1053.46262E-0828,879,866.3
1070.000000187288455,339,357.599.9999812712%1062.98642E-0833,484,921.9
1080.000000161531376,190,747.899.9999838469%1072.57571E-0838,824,279.5
1090.000000139316577,177,897.199.9999860683%1082.22148E-0845,015,027.1
1100.000000120156888,322,452.899.9999879843%1091.91597E-0852,192,924.4
1110.000000103632169,649,514.399.9999896368%1101.65247E-0860,515,377.1
1120.0000000893800211,188,183.199.9999910620%1111.42521E-0870,164,891.6
1130.0000000770879212,972,201.399.9999922912%1121.22921E-0881,353,074.4
1140.0000000664863115,040,691.199.9999933514%1131.06016E-0894,325,275.8
1150.0000000573426917,439,013.299.9999942657%1149.14361E-09109,365,966.7
1160.0000000494565720,219,761.199.9999950543%1157.88613E-09126,804,979.5
1170.0000000426549923,443,914.799.9999957345%1166.80158E-09147,024,741.8
1180.0000000367888127,182,177.699.9999963211%1175.86618E-09170,468,654.7
1190.0000000317293831,516,527.499.9999968271%1185.05943E-09197,650,832.5
1200.0000000273657636,542,013.499.9999972634%1194.36362E-09229,167,361.7
1210.0000000236022542,368,841.199.9999976398%1203.76351E-09265,709,376.6
1220.0000000203563249,124,789.199.9999979644%1213.24593E-09308,078,214.0
1230.0000000175567956,958,010.699.9999982443%1222.79953E-09357,203,007.8
1240.0000000151422766,040,282.999.9999984858%1232.41452E-09414,161,006.8
1250.0000000130598176,570,774.299.9999986940%1242.08246E-09480,201,304.5
rollsor more1 inor lessrollsrelative1 in
1260.0000000112637588,780,411.199.9999988736%1251.79607E-09556,772,080.0
1270.00000000971469102,936,942.699.9999990285%1261.54906E-09645,552,466.6
1280.00000000837866119,350,812.199.9999991621%1271.33602E-09748,489,447.2
1290.00000000722638138,381,964.599.9999992774%1281.15229E-09867,840,236.0
1300.00000000623256160,447,740.299.9999993767%1299.93816E-101,006,222,112.8
1310.00000000537542186,032,027.099.9999994625%1308.5714E-101,166,670,067.3
1320.00000000463616215,695,870.999.9999995364%1317.39261E-101,352,701,845.4
1330.00000000399856250,089,780.199.9999996001%1326.37593E-101,568,397,883.1
1340.00000000344866289,967,989.999.9999996551%1335.49908E-101,818,487,443.3
1350.00000000297438336,205,002.599.9999997026%1344.74281E-102,108,455,911.4
1360.00000000256532389,814,764.799.9999997435%1354.09055E-102,444,660,286.0
1370.00000000221252451,972,902.399.9999997787%1363.52799E-102,834,475,948.2
1380.00000000190824524,042,501.599.9999998092%1373.0428E-103,286,447,400.4
1390.00000000164581607,604,000.199.9999998354%1382.62433E-103,810,492,076.4
1400.00000000141947704,489,845.599.9999998581%1392.26342E-104,418,094,044.3
1410.00000000122425816,824,679.299.9999998776%1401.95214E-105,122,584,573.0
1420.00000000105589947,071,928.499.9999998944%1411.68367E-105,939,410,552.6
1430.000000000910671,098,087,827.799.9999999089%1421.45212E-106,886,480,147.4
1440.000000000785431,273,184,054.199.9999999215%1431.25242E-107,984,567,755.0
1450.000000000677411,476,200,350.199.9999999323%1441.08018E-109,257,751,028.6
1460.000000000584251,711,588,726.399.9999999416%1459.31623E-1110,733,960,793.7
1470.000000000503901,984,511,091.599.9999999496%1468.03501E-1112,445,541,431.6
1480.000000000434602,300,952,449.599.9999999565%1476.92998E-1114,430,058,546.8
1490.000000000374832,667,852,146.399.9999999625%1485.97693E-1116,730,997,178.0
1500.000000000323283,093,256,045.499.9999999677%1495.15494E-1119,398,856,069.4
rollsor more1 inor lessrollsrelative1 in
1510.000000000278823,586,492,968.099.9999999721%1504.446E-1122,492,132,184.8
1520.000000000240484,158,379,267.999.9999999760%1513.83457E-1126,078,570,573.7
1530.000000000207414,821,456,026.899.9999999793%1523.30721E-1130,236,966,550.2
1540.000000000178885,590,264,071.899.9999999821%1532.85237E-1135,058,517,488.9
1550.000000000154286,481,662,846.199.9999999846%1542.46011E-1140,648,590,642.7
1560.000000000133067,515,200,124.799.9999999867%1552.12177E-1147,130,474,563.3
1570.000000000114768,713,540,684.999.9999999885%1561.82998E-1154,645,387,700.9
1580.0000000000989810,102,963,328.699.9999999901%1571.5783E-1163,359,143,891.4
1590.0000000000853711,713,937,159.499.9999999915%1581.36124E-1173,462,191,132.4
1600.0000000000736313,581,789,749.499.9999999926%1591.17404E-1185,176,071,932.7
1610.0000000000635015,747,481,848.999.9999999936%1601.01258E-1198,757,735,373.5
1620.0000000000547718,258,505,628.399.9999999945%1618.73324E-12114,505,088,285.8
1630.0000000000472421,169,926,149.299.9999999953%1627.5322E-12132,763,387,399.6
1640.0000000000407424,545,588,904.699.9999999959%1636.49625E-12153,935,010,249.7
1650.0000000000351428,459,519,907.099.9999999965%1645.60285E-12178,480,546,402.4
1660.0000000000303132,997,549,029.499.9999999970%1654.83236E-12206,938,364,534.8
1670.0000000000261438,259,192,196.899.9999999974%1664.16778E-12239,936,048,341.5
1680.0000000000225444,359,833,703.099.9999999977%1673.59457E-12278,197,462,851.4
1690.0000000000194451,433,256,510.999.9999999981%1683.1003E-12322,549,659,972.8
1700.0000000000167759,634,576,022.799.9999999983%1692.67386E-12373,991,000,446.0
1710.0000000000144669,143,641,656.399.9999999986%1702.30616E-12433,622,147,830.8
1720.0000000000124780,168,980,822.099.9999999988%1711.98896E-12502,774,169,954.8
1730.0000000000107692,952,371,788.499.9999999989%1721.71552E-12582,914,784,800.7
1740.00000000000928107,774,145,717.099.9999999991%1731.47948E-12675,911,695,538.1
1750.00000000000800124,959,334,135.999.9999999992%1741.27609E-12783,643,575,321.1
rollsor more1 inor lessrollsrelative1 in
1760.00000000000690144,884,796,662.599.9999999993%1751.10056E-12908,624,962,649.1
1770.00000000000595167,987,485,281.499.9999999994%1769.49241E-131,053,473,597,045.7
1780.00000000000513194,774,026,407.499.9999999995%1778.18678E-131,221,480,777,697.5
1790.00000000000443225,831,830,861.899.9999999996%1787.05991E-131,416,449,010,023.7
1800.00000000000382261,841,975,396.199.9999999996%1796.09068E-131,641,851,851,028.3
1810.00000000000329303,594,138,247.599.9999999997%1805.25135E-131,904,270,455,547.8
1820.00000000000284352,003,916,250.799.9999999997%1814.53082E-132,207,105,918,829.0
1830.00000000000245408,132,903,260.599.9999999998%1823.90687E-132,559,590,581,057.4
1840.00000000000211473,211,970,190.799.9999999998%1833.36953E-132,967,775,701,726.9
1850.00000000000182548,668,257,185.099.9999999998%1842.90656E-133,440,488,638,174.6
1860.00000000000157636,156,469,839.799.9999999998%1852.50577E-133,990,783,896,650.9
1870.00000000000136737,595,165,784.299.9999999999%1862.1616E-134,626,193,762,065.2
1880.00000000000117855,208,827,358.799.9999999999%1871.86517E-135,361,428,127,822.0
1890.00000000000101991,576,643,014.799.9999999999%1881.6076E-136,220,441,474,268.6
1900.000000000000871,149,689,067,182.799.9999999999%1891.38778E-137,205,759,403,792.8
1910.000000000000751,333,013,398,924.699.9999999999%1901.19571E-138,363,230,505,794.8
1920.000000000000651,545,569,817,469.899.9999999999%1911.0314E-139,695,585,850,098.0
1930.000000000000561,792,019,542,039.899.9999999999%1928.90399E-1411,230,921,764,016.2
1940.000000000000482,077,767,049,248.8100.0000000000%1937.67164E-1413,035,020,629,147.6
1950.000000000000422,409,078,589,639.8100.0000000000%1946.61693E-1415,112,750,427,417.8
1960.000000000000362,793,219,602,341.5100.0000000000%1955.70655E-1417,523,733,958,640.1
1970.000000000000313,238,614,041,259.3100.0000000000%1964.92939E-1420,286,484,807,975.2
1980.000000000000273,755,029,106,716.0100.0000000000%1974.24105E-1423,579,055,640,683.2
1990.000000000000234,353,789,433,581.8100.0000000000%1983.66374E-1427,294,543,196,184.8
2000.000000000000205,048,025,432,896.4100.0000000000%1993.16414E-1431,604,207,911,371.9
2010.000000000000175,852,961,232,947.0100.0000000000%2002.72005E-1436,764,078,590,779.6
winsome johnny (not Win some johnny)
riverbed
riverbed
Joined: Jan 14, 2010
  • Threads: 4
  • Posts: 21
June 14th, 2012 at 12:36:19 PM permalink
Thank 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.

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?
7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
June 14th, 2012 at 1:14:53 PM permalink
Quote: riverbed

Thank 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.

Nice observations.
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.
Quote: riverbed

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?

No, the median is not 8.5 rolls
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
winsome johnny (not Win some johnny)
7craps
7craps
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June 14th, 2012 at 2:44:31 PM permalink
Quote: pacomartin

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.

The matrix algebra program does not have to be sophisticated.
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
winsome johnny (not Win some johnny)
guido111
guido111
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June 14th, 2012 at 3:19:16 PM permalink
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.

I thought I posted this here before, maybe at another forum.
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!
mustangsally
mustangsally
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June 14th, 2012 at 5:31:14 PM permalink
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silly
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riverbed
riverbed
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June 14th, 2012 at 6:45:33 PM permalink
Many thanks, 7craps. As a back line better, predominantly, this huge difference between the median (6) and the mean (8.5) is the crux of the game for me. I run a progression on the back line, and I'm standing there trying desperately to avoid those long rolls. Or at least recognize the hot shooter who is going to get my whole stake onto the back line before I switch to the front line. I'm constantly experimenting w/ switching over earlier in these long rolls so that my losses on individual shooters (those that 7out right after I have switched to the front line) are smaller (but more frequent) and my wins on the long rolls are bigger. Your chart illustrates so well where that 'ideal' switching-over point might be according to how hot/cold the situation seems to be.

Again, many thanks.
7craps
7craps
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June 14th, 2012 at 7:29:35 PM permalink
Quote: riverbed

Many thanks. But what accounts for the median and the average rolls per shooter being different?

The distribution. The probabilities of each possible outcome.

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
winsome johnny (not Win some johnny)
riverbed
riverbed
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June 15th, 2012 at 10:52:58 AM permalink
Yes, I get this now. The few long rolls skew the mean 40% above the median. Fully 60& of shooters have shorter rolls than average (the mean). This is a crucial description of the game. Your strategy had better accommodate all these short rolls! Your chart is highly instructive.

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