Ask the Wizard! No. 81

May 5, 2003

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 the 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

I hope you're happy. I think this is the longest answer to any “ask the wizard” question. It has been rewritten many times, with the last rewrite on June 1, 2009.

Yes, this happened at the California casino. There is a plaque in the side of the building mentioning it.

According to The Virgin Kiss by Frank Scoblete, Fujitake had 119 total rolls, including the seven out. The following table shows the probability of being in any of the four possible states in craps before the n^th roll. The states are defined as follows:

State 1 = Come out roll
State 2 = Point of 4 or 10
State 3 = Point of 5 or 9
State 4 = Point of 6 or 8

The probabilities are recursive. Let p(x,r) represent the probability of being in state x before roll r. Based on simple dice probabilities, p(x,r) can be expressed as follows:

(1) P(1,r) = (12/36)×p(1,r-1) + (3/36)×p(2,r-1) + (4/36)×p(3,r-1) + (5/36)× p(4,r-1)
(2) P(2,r) = (6/36)×p(1,r-1) + (27/36)×p(2,r-1)
(3) P(3,r) = (8/36)×p(1,r-1) + (26/36)×p(3,r-1)
(4) P(4,r) = (10/36)×p(1,r-1) + (25/36)×p(4,r-1)

In more plain simple English, equation (1) is saying that the probability of being in a come out roll before roll r is the sum of the following:

1. Product of the probability of being in a come out roll the previous turn, and the probability of staying in a come out roll (12/36).
2. Product of the probability of rolling for a 4 or 10 the previous roll, and the probability of making the point (3/36), resulting in a come out roll.
3. Product of the probability of rolling for a 5 or 9 the previous roll, and the probability of making the point (4/36), resulting in a come out roll.
4. Product of the probability of rolling for a 6 or 8 the previous roll, and the probability of making the point (5/36), resulting in a come out roll.

Equation (2) is saying the probability of rolling for a 4 or 10 before roll r is the sum of:

1. Product of the probability of being in a come out roll the previous turn, and the probability of rolling a 4 or 10 (6/36).
2. Product of the probability of rolling for a point of 4 or 10 the previous turn, and the probability of not rolling the desired point or a 7 (27/36).

Equation (3) is saying the probability of rolling for a 5 or 9 before roll r is the sum of:

1. Product of the probability of being in a come out roll the previous turn, and the probability of rolling a 5 or 9 (8/36).
2. Product of the probability of rolling for a point of 5 or 9 the previous turn, and the probability of not rolling the desired point or a 7 (26/36).

Equation (4) is saying the probability of rolling for a 6 or 8 before roll r is the sum of:

1. Product of the probability of being in a come out roll the previous turn, and the probability of rolling a 6 or 8 (10/36).
2. Product of the probability of rolling for a point of 6 or 8 the previous turn, and the probability of not rolling the desired point or a 7 (25/36).

That said, we can use a spreadsheet to recursively calculate the probability of each state for roll r, based on the probabilities for the previous state, r-1. We know that for roll 1 the probability is 1 for state 1, and 0 for the rest, because a shooter always starts with a come out roll. The following table shows the probabilities of each state for rolls 1 to 200.

Probability States in Craps — Recursive Table
Roll	State 1	State 2	State 3	State 4	Total	Inverse of Total
1	1	0	0	0	1	1
2	0.333333333333333	0.166666666666667	0.222222222222222	0.277777777777778	1	1
3	0.188271604938272	0.180555555555556	0.234567901234568	0.285493827160494	0.888888888888889	1.13
4	0.143518518518518	0.166795267489712	0.211248285322359	0.250557270233196	0.772119341563786	1.3
5	0.120010764365188	0.149016203703704	0.184461210181375	0.213864359472641	0.667352537722908	1.5
6	0.10262067837728	0.131763946838642	0.159891043878813	0.18185323973522	0.576128908829955	1.74
7	0.0882102876454065	0.115926406525195	0.138281460218538	0.154792716032036	0.497210870421176	2.01
8	0.0759275581209665	0.101646519501464	0.11947222963459	0.131997799368193	0.429044106625214	2.33
9	0.0653874492037588	0.0888894826462592	0.103158289874085	0.112756126817069	0.370191348541173	2.7
10	0.0563258786657103	0.0775650201853209	0.089033753621008	0.0964660461795646	0.319390698651604	3.13
11	0.0485297458312873	0.0675614115832757	0.0768190173186636	0.0826363872540616	0.275546561987288	3.63
12	0.041819421951951	0.058759349659338	0.0662647893593209	0.0708668649906781	0.237710425961288	4.21
13	0.0360417942997806	0.051039415903162	0.0571511083043875	0.0608296067857351	0.205061925293065	4.88
14	0.0310658957348077	0.0442865276440016	0.0492850880642311	0.0522543920178106	0.176891903460851	5.65
15	0.0267795178915465	0.0383925450221358	0.0424983182096798	0.0449171877164818	0.152587568839844	6.55
16	0.0230864183662804	0.0332576617485263	0.0366442337940013	0.038631246439653	0.131619560348461	7.6
17	0.0199039770282004	0.0287909827057748	0.031595595154841	0.0332401484626147	0.113530703351431	8.81
18	0.0171612165385602	0.0249105665340311	0.0272421469514297	0.0286123189402048	0.0979262489642258	10.2
19	0.0147971244603211	0.0215431276569501	0.0234884875846015	0.0246366705247422	0.0844654102266148	11.8
20	0.0127592272071339	0.0186235331527661	0.0202521575800613	0.0212191113256046	0.0728540292655659	13.7
21	0.0110023753581159	0.0160941877324302	0.017461942076074	0.0182797237558737	0.0628382289224939	15.9
22	0.0094877070716207	0.0139043700256753	0.0150563749123014	0.0157504679855001	0.0541989199950976	18.5
23	0.0081817620698106	0.0120095620311933	0.0129824278970223	0.0135732991764919	0.0467470511745181	21.4
24	0.0070557232889404	0.01037079853503	0.011194367274474	0.0116986138919557	0.0403195029904002	24.8
25	0.0060847670452792	0.0089540527827626	0.0096527593179958	0.010083960560786	0.0347755397068235	28.8
26	0.0052475056379875	0.0077296674279518	0.0083236077397257	0.0086929634575678	0.0299937442632328	33.3
27	0.0045255088385124	0.0066718348439618	0.007177606842688	0.0074944206338631	0.0258693711590253	38.7
28	0.0039028928092806	0.0057581276060567	0.0061894957949441	0.0064615445619917	0.0223120607722731	44.8
29	0.0033659667366465	0.0049690778394227	0.0053375009206331	0.0055713206150722	0.0192438661117744	52
30	0.0029029289198866	0.0042878028356747	0.0046028543841564	0.0048039634095353	0.016597549549253	60.2
31	0.0025036052813879	0.0036996736134038	0.0039693790374211	0.0041424548454791	0.014315112777692	69.9
32	0.0021592242942763	0.0031920227569509	0.0034231304784459	0.0035721506653016	0.0123465281949746	81
33	0.0018622231956239	0.0027538877834259	0.0029520885220501	0.0030804447104251	0.0106486442115249	93.9
34	0.0016060810927247	0.0023757863701734	0.0025458913093971	0.0026564819365796	0.0091842407088747	109
35	0.0013851751984362	0.0020495199597508	0.0021956061885034	0.0022909127594927	0.0079212141061831	126
36	0.001194656966999	0.0017680025028858	0.0018935322913494	0.0019756825269911	0.0068318742882253	146
37	0.0010303453586944	0.0015251113716642	0.0016330304253077	0.0017038509123547	0.0058923380680209	170
38	0.0008886348522869	0.0013155577551972	0.0014083764979876	0.0014694368443281	0.0050820059497998	197
39	0.0007664161585174	0.0011347741251124	0.0012146352157215	0.0012672852675297	0.004383110766881	228
40	0.0006610078743911	0.0009788166202539	0.0010475512465805	0.0010929525909283	0.0037803283321537	265
41	0.0005700975637338	0.0008442804442556	0.0009034554279506	0.0009426081532533	0.0032604415891933	307
42	0.0004916909604345	0.000728226593814	0.0007791839343496	0.0008129494296853	0.0028120509182834	356
43	0.0004240681720134	0.0006281184387662	0.0006720086104602	0.0007011290374021	0.0024253242586419	412
44	0.0003657459169253	0.0005417668577436	0.0005795769235575	0.0006046918793107	0.0020917815775371	478
45	0.0003154449629755	0.0004672827961286	0.0004998602041083	0.0005215210042228	0.0018041089674352	554
46	0.00027206204949	0.0004030362575923	0.0004311090280728	0.0004497909648702	0.0015559983000252	643
47	0.0002346456750916	0.0003476208681092	0.0003718147534948	0.0003879276282404	0.001342008924936	745
48	0.0002023752183502	0.0002998232635972	0.0003206763608777	0.0003345735404702	0.0011574483832953	864
49	0.0001745429321349	0.0002585966507563	0.0002765718647117	0.0002885580748682	0.0009982695224711	1002
50	0.0001505384158632	0.0002230379767564	0.0002385336649884	0.0002488716998071	0.0008609817574151	1161
51	0.0001298352244338	0.0001923682185445	0.0002057272949057	0.0002146437959391	0.0007425745338231	1347
52	0.000111979319671	0.0001659153679807	0.000177433096195	0.0001851235317449	0.0006404513155915	1561
53	0.0000965791106527	0.0001430997459307	0.0001530304182899	0.0001596633747315	0.0005523726496048	1810
54	0.0000832968642345	0.0001234213278901	0.0001319839933544	0.0001377048743004	0.0004764070597795	2099
55	0.0000718412972057	0.0001064488066233	0.0001138321872525	0.0001187664027737	0.0004108886938553	2434
56	0.0000619611874782	0.0000918101545018	0.0000981768679503	0.0001024325844833	0.0003543807944137	2822
57	0.0000534398640962	0.000079184480456	0.0000846746685148	0.0000883451801907	0.0003056441932578	3272
58	0.0000460904551538	0.0000682950043581	0.0000730294526154	0.0000761952262703	0.0002636101383975	3793
59	0.0000397517893537	0.0000589029957942	0.0000629858169231	0.000065716255786	0.0002273568578569	4398
60	0.0000342848612847	0.0000508025450712	0.0000543234876342	0.0000566784524496	0.0001960893464397	5100
61	0.0000295697828726	0.0000438160523509	0.0000468524880213	0.0000488836090024	0.0001691219322472	5913
62	0.0000255031541284	0.0000377903364086	0.0000404089708759	0.0000421607792719	0.0001458632406848	6856
63	0.0000219957955175	0.0000325932779945	0.0000348516243278	0.0000363625284189	0.0001258032262587	7949
64	0.0000189707922111	0.0000281109244155	0.0000300585721295	0.0000313616990457	0.0001085019878018	9216
65	0.0000163618073202	0.0000242449920134	0.0000259247003627	0.0000270486221737	0.00009358012187	10686
66	0.0000141116271167	0.0000209107118968	0.0000223593518886	0.0000233287118763	0.0000807104027784	12390
67	0.0000121709063341	0.0000180349717754	0.0000192843379455	0.0000201203907798	0.0000696106068348	14366
68	0.0000104970860282	0.0000155547132205	0.0000166322232571	0.0000173533009121	0.000060037323418	16656
69	0.0000090534602664	0.0000134155492534	0.0000143448470253	0.0000149667606413	0.0000517806171864	19312
70	0.000007808371174	0.0000115705719845	0.0000123720473552	0.0000129084338526	0.0000446594243664	22392
71	0.0000067345146868	0.000009979324184	0.0000106705611286	0.0000111331821682	0.0000385175821676	25962
72	0.0000058083417819	0.0000086069122525	0.0000092030751899	0.0000096020750298	0.0000332204042542	30102
73	0.0000050095420569	0.000007423241153	0.0000079374080332	0.0000082815359324	0.0000286517271755	34902
74	0.0000043205983316	0.0000064023545409	0.0000068458040366	0.00000714260608	0.000024711362989	40467
75	0.0000037264025041	0.0000055218656276	0.0000059043247668	0.0000061603093143	0.0000213129022128	46920
76	0.0000032139242381	0.0000047624663047	0.0000050923239991	0.0000053131043861	0.000018381818928	54402
77	0.0000027719252138	0.0000041075037682	0.0000043919949412	0.0000045824125565	0.0000158538364797	63076
78	0.0000023907126783	0.0000035426153618	0.0000037879797273	0.000003952210168	0.0000136735179354	73134
79	0.0000020619268882	0.0000030554136344	0.0000032670326204	0.0000034086772496	0.0000117930503925	84796
80	0.0000017783577858	0.0000026352147072	0.0000028177295343	0.0000029398944478	0.0000101711964751	98317
81	0.0000015337868869	0.0000022728039947	0.000002430217505	0.0000025355816404	0.0000087723900269	113994
82	0.0000013228509125	0.0000019602341438	0.0000020959986174	0.0000021868724966	0.0000075659561702	132171
83	0.0000011409241759	0.0000016906507599	0.0000018077436486	0.0000018861200428	0.0000065254386273	153246
84	0.0000009840171444	0.0000014581420993	0.0000015591313409	0.0000016267289675	0.0000056280195521	177682
85	0.0000008486889509	0.0000012576094319	0.0000013447097783	0.0000014030109897	0.0000048540191508	206015
86	0.000000731971938	0.0000010846552324	0.000001159776829	0.0000012100601181	0.0000041864641175	238865
87	0.0000006313065794	0.0000009354867473	0.0000010002770294	0.0000010436450648	0.0000036107154209	276953
88	0.000000544485351	0.0000008068328237	0.00000086271265	0.000000900116456	0.0000031141472806	321115
89	0.0000004696043212	0.0000006958721763	0.0000007440669919	0.000000776326803	0.0000026858702924	372319
90	0.0000004050213991	0.0000006001715191	0.0000006417382322	0.0000006695614802	0.0000023164926305	431687
91	0.0000003493203243	0.0000005176322058	0.0000005534823675	0.0000005774791943	0.0000019979140919	500522
92	0.0000003012796097	0.0000004464442084	0.0000004773640041	0.0000004980606417	0.000001723148464	580333
93	0.0000002598457546	0.0000003850464246	0.0000004117139163	0.0000004295642261	0.0000014861703216	672870
94	0.0000002241101424	0.0000003320924442	0.0000003550924406	0.0000003704878666	0.0000012817828938	780163
95	0.0000001932891149	0.0000002864210236	0.0000003062579054	0.0000003195360581	0.0000011055041019	904565
96	0.0000001667067878	0.0000002470306202	0.0000002641394016	0.0000002755914611	0.0000009534682707	1048803
97	0.0000001437802285	0.0000002130574298	0.0000002278132985	0.0000002376904002	0.0000008223413569	1216040
98	0.0000001240066729	0.0000001837564437	0.0000001964829886	0.0000002050017303	0.0000007092478355	1409944
99	0.0000001069525003	0.0000001584851116	0.0000001694614191	0.0000001768086107	0.0000006117076417	1634768
100	0.0000000922437241	0.0000001366892504	0.000000146156025	0.0000001524927853	0.0000005275817848	1895441
101	0.0000000795577908	0.0000001178908919	0.0000001260557345	0.0000001315210243	0.0000004550254414	2197679
102	0.0000000686165062	0.0000001016778007	0.0000001087197617	0.000000113433431	0.0000003924474996	2548112
103	0.0000000591799355	0.0000000876944349	0.0000000937679404	0.0000000978333566	0.0000003384756674	2954422
104	0.000000051041141	0.0000000756341487	0.0000000808723871	0.0000000843787019	0.0000002919263787	3425521
105	0.0000000440216443	0.0000000652324684	0.0000000697503109	0.0000000727744155	0.0000002517788391	3971740
106	0.0000000379675127	0.000000056261292	0.0000000601578122	0.0000000627660231	0.00000021715264	4605056
107	0.0000000327459831	0.0000000485238878	0.0000000518845338	0.0000000541340473	0.0000001872884521	5339357
108	0.0000000282425509	0.0000000418505797	0.0000000447490485	0.0000000466891949	0.0000001615313739	6190748
109	0.0000000243584588	0.0000000360950266	0.0000000385948797	0.000000040268205	0.0000001393165701	7177897
110	0.0000000210085314	0.0000000311310131	0.0000000332870706	0.0000000347302698	0.0000001201568849	8322453
111	0.0000000181193069	0.0000000268496817	0.0000000287092246	0.0000000299539461	0.0000001036321593	9649514
112	0.0000000156274266	0.0000000231571458	0.0000000247609526	0.0000000258344923	0.0000000893800172	11188183
113	0.0000000134782452	0.0000000199724304	0.0000000213556717	0.0000000222815714	0.0000000770879188	12972201
114	0.0000000116246327	0.000000017225697	0.0000000184187063	0.0000000192172705	0.0000000664863065	15040691
115	0.0000000100259406	0.0000000148567115	0.0000000158856507	0.0000000165743914	0.0000000573426942	17439013
116	0.0000000086471106	0.0000000128135238	0.0000000137009567	0.0000000142949775	0.0000000494565686	20219761
117	0.0000000074579059	0.0000000110513279	0.0000000118167156	0.0000000123290429	0.0000000426549923	23443915
118	0.0000000064322481	0.0000000095314803	0.000000010191607	0.0000000106334759	0.0000000367888112	27182178
119	0.0000000055476451	0.0000000082206515	0.0000000087899935	0.0000000091710938	0.000000031729384	31516527
120	0.0000000047846983	0.0000000070900962	0.0000000075811387	0.0000000079098277	0.0000000273657609	36542013
121	0.0000000041266767	0.0000000061150219	0.0000000065385331	0.0000000068220188	0.0000000236022505	42368841
122	0.0000000035591503	0.0000000052740458	0.0000000056393132	0.0000000058838121	0.0000000203563215	49124789
123	0.0000000030696738	0.0000000045487261	0.0000000048637596	0.0000000050746335	0.000000017556793	56958011
124	0.000000002647513	0.0000000039231569	0.000000004194865	0.0000000043767382	0.0000000151422731	66040283
125	0.0000000022834105	0.0000000033836198	0.0000000036179609	0.0000000037748218	0.0000000130598131	76570774
126	0.0000000019693816	0.0000000029182833	0.0000000031203964	0.0000000032556847	0.000000011263746	88780411
127	0.00000000169854	0.0000000025169427	0.0000000026912599	0.0000000028079426	0.0000000097146853	102936943
128	0.0000000014649461	0.000000002170797	0.0000000023211411	0.0000000024217768	0.000000008378661	119350812
129	0.0000000012634776	0.0000000018722555	0.0000000020019232	0.0000000020887189	0.0000000072263752	138381964
130	0.0000000010897163	0.0000000016147712	0.0000000017266062	0.0000000018014653	0.000000006232559	160447740
131	0.0000000009398517	0.0000000013926978	0.0000000014891526	0.0000000015537165	0.0000000053754185	186032027
132	0.0000000008105974	0.0000000012011653	0.000000001284355	0.0000000013400397	0.0000000046361574	215695871
133	0.000000000699119	0.0000000010359735	0.0000000011077225	0.0000000011557491	0.000000003998564	250089780
134	0.0000000006029718	0.0000000008935	0.0000000009553816	0.0000000009968032	0.0000000034486565	289967990
135	0.0000000005200473	0.0000000007706203	0.0000000008239915	0.0000000008597166	0.0000000029743757	336205003
136	0.0000000004485272	0.0000000006646398	0.0000000007106711	0.000000000741483	0.000000002565321	389814765
137	0.0000000003868429	0.0000000005732343	0.0000000006129351	0.0000000006395096	0.000000002212522	451972902
138	0.0000000003336419	0.0000000004943996	0.0000000005286405	0.0000000005515603	0.0000000019082422	524042501
139	0.0000000002877573	0.0000000004264067	0.0000000004559385	0.0000000004757063	0.0000000016458088	607604000
140	0.0000000002481832	0.0000000003677645	0.000000000393235	0.0000000004102842	0.0000000014194669	704489846
141	0.0000000002140515	0.0000000003171873	0.0000000003391549	0.0000000003538593	0.0000000012242529	816824679
142	0.0000000001846138	0.0000000002735657	0.0000000002925122	0.0000000003051944	0.000000001055886	947071928
143	0.0000000001592245	0.0000000002359432	0.0000000002522841	0.0000000002632221	0.000000000910674	1098087828
144	0.000000000137327	0.0000000002034948	0.0000000002175884	0.0000000002270222	0.0000000007854324	1273184054
145	0.0000000001184409	0.000000000175509	0.0000000001876643	0.0000000001958007	0.0000000006774148	1476200350
146	0.0000000001021522	0.0000000001513719	0.0000000001618555	0.0000000001688729	0.0000000005842525	1711588726
147	0.0000000000881036	0.0000000001305543	0.0000000001395961	0.0000000001456485	0.0000000005039024	1984511092
148	0.000000000075987	0.0000000001125996	0.000000000120398	0.000000000125618	0.0000000004346026	2300952450
149	0.0000000000655368	0.0000000000971142	0.0000000001038401	0.0000000001083422	0.0000000003748334	2667852146
150	0.0000000000565238	0.0000000000837585	0.0000000000895594	0.0000000000934423	0.0000000003232839	3093256045
151	0.0000000000487503	0.0000000000722395	0.0000000000772426	0.0000000000805916	0.0000000002788239	3586492968
152	0.0000000000420458	0.0000000000623047	0.0000000000666197	0.0000000000695081	0.0000000002404783	4158379268
153	0.0000000000362634	0.0000000000537361	0.0000000000574578	0.0000000000599489	0.0000000002074062	4821456027
154	0.0000000000312763	0.000000000046346	0.0000000000495558	0.0000000000517044	0.0000000001788824	5590264072
155	0.0000000000269749	0.0000000000399722	0.0000000000427406	0.0000000000445937	0.0000000001542814	6481662846
156	0.0000000000232652	0.000000000034475	0.0000000000368626	0.0000000000384609	0.0000000001330637	7515200125
157	0.0000000000200656	0.0000000000297338	0.0000000000317931	0.0000000000331715	0.0000000001147639	8713540685
158	0.0000000000173061	0.0000000000256446	0.0000000000274207	0.0000000000286095	0.0000000000989809	10102963329
159	0.000000000014926	0.0000000000221178	0.0000000000236496	0.000000000024675	0.0000000000853684	11713937159
160	0.0000000000128733	0.000000000019076	0.0000000000203972	0.0000000000212815	0.000000000073628	13581789749
161	0.0000000000111029	0.0000000000164526	0.000000000017592	0.0000000000183547	0.0000000000635022	15747481849
162	0.0000000000095759	0.0000000000141899	0.0000000000151727	0.0000000000158305	0.000000000054769	18258505628
163	0.000000000008259	0.0000000000122384	0.000000000013086	0.0000000000136534	0.0000000000472368	21169926149
164	0.0000000000071232	0.0000000000105553	0.0000000000112863	0.0000000000117757	0.0000000000407405	24545588905
165	0.0000000000061436	0.0000000000091037	0.0000000000097342	0.0000000000101562	0.0000000000351376	28459519907
166	0.0000000000052987	0.0000000000078517	0.0000000000083955	0.0000000000087595	0.0000000000303053	32997549029
167	0.0000000000045699	0.0000000000067719	0.0000000000072409	0.0000000000075548	0.0000000000261375	38259192197
168	0.0000000000039415	0.0000000000058406	0.0000000000062451	0.0000000000065158	0.0000000000225429	44359833703
169	0.0000000000033994	0.0000000000050373	0.0000000000053862	0.0000000000056197	0.0000000000194427	51433256511
170	0.0000000000029319	0.0000000000043446	0.0000000000046455	0.0000000000048469	0.0000000000167688	59634576023
171	0.0000000000025287	0.0000000000037471	0.0000000000040066	0.0000000000041803	0.0000000000144626	69143641656
172	0.0000000000021809	0.0000000000032318	0.0000000000034556	0.0000000000036054	0.0000000000124737	80168980822
173	0.000000000001881	0.0000000000027873	0.0000000000029803	0.0000000000031096	0.0000000000107582	92952371788
174	0.0000000000016223	0.000000000002404	0.0000000000025705	0.0000000000026819	0.0000000000092787	107774145717
175	0.0000000000013992	0.0000000000020734	0.000000000002217	0.0000000000023131	0.0000000000080026	124959334136
176	0.0000000000012068	0.0000000000017882	0.0000000000019121	0.000000000001995	0.000000000006902	144884796662
177	0.0000000000010408	0.0000000000015423	0.0000000000016491	0.0000000000017206	0.0000000000059528	167987485281
178	0.0000000000008977	0.0000000000013302	0.0000000000014223	0.000000000001484	0.0000000000051342	194774026407
179	0.0000000000007742	0.0000000000011473	0.0000000000012267	0.0000000000012799	0.0000000000044281	225831830862
180	0.0000000000006677	0.0000000000009895	0.000000000001058	0.0000000000011039	0.0000000000038191	261841975396
181	0.0000000000005759	0.0000000000008534	0.0000000000009125	0.0000000000009521	0.0000000000032939	303594138248
182	0.0000000000004967	0.000000000000736	0.000000000000787	0.0000000000008211	0.0000000000028409	352003916251
183	0.0000000000004284	0.0000000000006348	0.0000000000006788	0.0000000000007082	0.0000000000024502	408132903260
184	0.0000000000003695	0.0000000000005475	0.0000000000005854	0.0000000000006108	0.0000000000021132	473211970191
185	0.0000000000003187	0.0000000000004722	0.0000000000005049	0.0000000000005268	0.0000000000018226	548668257185
186	0.0000000000002748	0.0000000000004073	0.0000000000004355	0.0000000000004544	0.0000000000015719	636156469840
187	0.000000000000237	0.0000000000003513	0.0000000000003756	0.0000000000003919	0.0000000000013558	737595165784
188	0.0000000000002044	0.000000000000303	0.0000000000003239	0.000000000000338	0.0000000000011693	855208827359
189	0.0000000000001763	0.0000000000002613	0.0000000000002794	0.0000000000002915	0.0000000000010085	991576643015
190	0.0000000000001521	0.0000000000002254	0.000000000000241	0.0000000000002514	0.0000000000008698	1149689067183
191	0.0000000000001312	0.0000000000001944	0.0000000000002078	0.0000000000002168	0.0000000000007502	1333013398925
192	0.0000000000001131	0.0000000000001676	0.0000000000001792	0.000000000000187	0.000000000000647	1545569817470
193	0.0000000000000976	0.0000000000001446	0.0000000000001546	0.0000000000001613	0.000000000000558	1792019542040
194	0.0000000000000841	0.0000000000001247	0.0000000000001333	0.0000000000001391	0.0000000000004813	2077767049249
195	0.0000000000000726	0.0000000000001075	0.000000000000115	0.00000000000012	0.0000000000004151	2409078589640
196	0.0000000000000626	0.0000000000000928	0.0000000000000992	0.0000000000001035	0.000000000000358	2793219602342
197	0.000000000000054	0.00000000000008	0.0000000000000855	0.0000000000000892	0.0000000000003088	3238614041259
198	0.0000000000000466	0.000000000000069	0.0000000000000738	0.000000000000077	0.0000000000002663	3755029106716
199	0.0000000000000402	0.0000000000000595	0.0000000000000636	0.0000000000000664	0.0000000000002297	4353789433582
200	0.0000000000000346	0.0000000000000513	0.0000000000000549	0.0000000000000573	0.0000000000001981	5048025432896

To verify this table, I did a random simulation of over 21 billion new shooters. The following table shows the count for each number of throws. The average rolls per shooter was 8.53.

Total Rolls in Craps Simulation
Rolls	Count	Probability
2	2337166954	0.111113765998
3	2456226208	0.116774089950
4	2203680193	0.104767528430
5	1918801263	0.091223793049
6	1659906169	0.078915383142
7	1433725096	0.068162265665
8	1237906639	0.058852649948
9	1068528927	0.050800082105
10	922222110	0.043844352477
11	795811607	0.037834534896
12	686746351	0.032649346344
13	592513250	0.028169309214
14	511244058	0.024305603214
15	441056957	0.020968762813
16	380450121	0.018087388086
17	328223382	0.015604420557
18	283126061	0.013460400352
19	244241934	0.01161176828
20	210694371	0.010016847533
21	181697608	0.008638281259
22	156748704	0.0074521586
23	135198981	0.006427640059
24	116623267	0.005544512076
25	100580892	0.004781824284
26	86732484	0.004123442236
27	74831327	0.003557636541
28	64530253	0.003067902111
29	55658630	0.002646126747
30	48000095	0.002282024104
31	41403022	0.001968385566
32	35715570	0.001697992298
33	30804938	0.001464530665
34	26567244	0.001263061900
35	22912522	0.001089308833
36	19764940	0.000939666255
37	17049802	0.000810582961
38	14701128	0.000698922126
39	12682839	0.000602968480
40	10936628	0.000519949986
41	9430433	0.000448342350
42	8135964	0.000386800609
43	7015476	0.000333530284
44	6049888	0.000287624227
45	5220048	0.000248171912
46	4501337	0.000214002900
47	3881712	0.000184544642
48	3347205	0.00015913307
49	2887420	0.000137273937
50	2492557	0.000118501331
51	2150924	0.00010225939
52	1851857	0.000088041124
53	1599229	0.000076030665
54	1378541	0.000065538699
55	1189435	0.000056548208
56	1027756	0.000048861653
57	883491	0.000042002995
58	763197	0.000036283969
59	658592	0.00003131083
60	566426	0.000026929067
61	488581	0.000023228154
62	421952	0.000020060474
63	363167	0.000017265713
64	312455	0.000014854759
65	271042	0.000012885899
66	234287	0.00001113849
67	200619	0.000009537843
68	173758	0.000008260816
69	149132	0.000007090045
70	129798	0.000006170866
71	111607	0.000005306028
72	95569	0.000004543549
73	83244	0.000003957592
74	71524	0.000003400399
75	61671	0.000002931967
76	53346	0.000002536180
77	45909	0.000002182609
78	39773	0.000001890891
79	34436	0.000001637159
80	29357	0.000001395693
81	25364	0.000001205857
82	21759	0.000001034468
83	18926	0.000000899781
84	16250	0.000000772559
85	14085	0.00000066963
86	12153	0.000000577779
87	10348	0.000000491965
88	9106	0.000000432918
89	7727	0.000000367358
90	6824	0.000000324427
91	5846	0.000000277931
92	5008	0.000000238091
93	4244	0.000000201769
94	3730	0.000000177332
95	3144	0.000000149472
96	2868	0.000000136351
97	2378	0.000000113055
98	2076	0.000000098697
99	1739	0.000000082676
100	1552	0.000000073785
101	1346	0.000000063992
102	1150	0.000000054673
103	950	0.000000045165
104	841	0.000000039983
105	778	0.000000036988
106	618	0.000000029381
107	541	0.00000002572
108	441	0.000000020966
109	412	0.000000019587
110	361	0.000000017163
111	298	0.000000014168
112	259	0.000000012313
113	201	0.000000009556
114	188	0.000000008938
115	160	0.000000007607
116	160	0.000000007607
117	123	0.000000005848
118	122	0.000000005800
119	85	0.000000004041
120	62	0.000000002948
121	70	0.000000003328
122	60	0.000000002853
123	47	0.000000002234
124	34	0.000000001616
125	50	0.000000002377
126	36	0.000000001712
127	41	0.000000001949
128	30	0.000000001426
129	17	0.000000000808
130	16	0.000000000761
131	9	0.000000000428
132	14	0.000000000666
133	11	0.000000000523
134	17	0.000000000808
135	14	0.000000000666
136	8	0.00000000038
137	7	0.000000000333
138	7	0.000000000333
139	3	0.000000000143
140	5	0.000000000238
141	8	0.00000000038
142	4	0.00000000019
143	2	0.000000000095
144	2	0.000000000095
145	2	0.000000000095
146	4	0.000000000190
147	2	0.000000000095
148	3	0.000000000143
149	2	0.000000000095
150	2	0.000000000095
151	2	0.000000000095
152	1	0.000000000048
153	1	0.000000000048
155	1	0.000000000048
160	1	0.000000000048
161	1	0.000000000048
165	1	0.000000000048
166	1	0.000000000048
172	1	0.000000000048
Total	21034000000	1

It seems to me that the simulation confirms my recursive results in the first table.

According to the recursive table, the probability of Mr. Fujitake making it to the 119th roll is 1 in 31.5 million.

In 'The Virgin Kiss', Scoblete tells the story of a skilled dice thrower, by the handle of "the Captain," making 148 throws in Atlantic City (including the seven out), in July 2005. The exact date and casino are not given, that I can find. According to the recursive table, the probability of making it to the 148th roll, assuming random dice throws, is 1 in 2.3 billion.

As far as I know, the longest confirmed roll is by Patricia Demauro, set at the Borgata in Atlantic City on May 23, 2009 (source). Patricia held the dice for 4 hours and 20 minutes, for a total of 154 rolls. According to the recursive table, the probability of making it to the 154th roll is 1 in 5.6 billion.

Also see my solution expressed in matrices at mathproblems.info, problem 204.

My thanks to BruceZ for his help with this question.

Let's say you have a $10 don't pass bet and the point is a 4. You have a 2/3 chance of winning the bet, so the expected value is (2/3)*$10 + (1/3)*-$10 =$ 10/3 = $3.33. Now consider adding a $40 odds bet on top of it. Now you have a 2/3 chance of winning $30 and a 1/3 chance of losing $50. The expected value of both bets combined is (2/3)*$30 + (1/3)*-$50 = $10/3 = $3.33. So either way your expected gain is 3 dollars and 33 cents. With the don't pass alone the player edge is $3.33/$10 = 33.33%. With the don't pass and odds the player edge is $3.33/$50 = 6.67%. So, yes, the player edge as a percentage drops by making the odds bet. However that player edge is effective over more money. The way I think gamblers should view the house edge is as the price to pay for entertainment. If you want to pay as little as possible then taking or laying the odds is getting entertainment for free.

It took me years to get the splitting pairs correct myself. Cindy of Gambling Tools was very helpful. Peter Griffin also addresses this topic in chapter 11 of the The Theory of Blackjack Let's say I want to determine the expected value of splitting eights against a dealer 2. Resplitting up to four hands is allowed. Here is how I did it.

1. Take a 2 and two 8's out of the shoe.
2. Determine the probability that the player will not get a third eight on either hand.
3. Go through all ranks, except 8, subtract that card from the deck, play out a hand with that card and an 8, determine the expected value, and multiply by 2. For each rank determine the probability of that rank, given that the probability of another 8 is zero. Take the dot product of the probability and expected value over each rank.
4. Multiply this dot product by the probability from step 2.
5. Determine the probability that the player will resplit to 3 hands.
6. Take another 8 out of the deck.
7. Repeat step 3 but multiply by 3 instead of 2.
8. Multiply dot product from step 7 by probability in step 5.
9. Determine the probability that the player will resplit to 4 hands.
10. Take two more 8's out of the shoe.
11. Repeat step 3 but multiply by 4 instead of 2, and this time consider getting an 8 as a third card, corresponding to the situation where the player is forced to stop resplitting.
12. Multiply dot product from step 11 by probability in step 9.
13. Add values from steps 4, 8, and 12.

The hardest part of all this is step 3. I have a very ugly subroutine full of long formulas I determine using probability trees. It gets especially ugly when the dealer has a 10 or ace up.

The formulaic answer for the number of combinations would be combin(8,2)*combin(6,2)*combin(4,2)/fact(4) = 25*15*6/24 = 105. Another way to solve the number of combinations would be to take one golfer at random. There are 7 possible people to pair him with. Then pick another golfer at random from the six left. There are 5 possible people to pair him with. Then pick another golfer at random from the four left. There are 3 possible people to pair him with. So the number of combination is 7*5*3 = 105. Thus the answer is 1 in 105.

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