Rullataulukon lukumäärä crapsissa

Johdanto

Yksi kysymys, jota minulta kysytään usein, on "mikä on todennäköisyys, että heittäjä selviää x heittoa crapsissa?" Seuraava taulukko vastaa tähän kysymykseen jopa 50 heitolle. Ensimmäinen sarake on heiton numero. Toinen sarake on todennäköisyys sille, että juuri kyseisellä heitolla tulee seitsemän ulos. Kolmas sarake on todennäköisyys selvitä kyseisen heiton JÄLKEEN.

Crapsin selviytymistaulukko 1–50 rullan kierrosten osalta

Rulla Määrä	Todennäköisyys Seitsemän ulos	Todennäköisyys Selviytyminen
1	0.00000000	1.00000000
2	0.11111111	0.88888889
3	0.11676955	0.77211934
4	0.10476680	0.66735254
5	0.09122363	0,57612891
6	0.07891804	0.49721087
7	0,06816676	0.42904411
8	0,05885276	0.37019135
9	0.05080065	0.31939070
10	0.04384414	0,27554656
11	0.03783614	0.23771043
12	0,03264850	0.20506193
13	0,02817002	0.17689190
14	0.02430433	0.15258757
15	0.02096801	0.13161956
16	0.01808886	0.11353070
17	0.01560445	0,09792625
18	0.01346084	0.08446541
19	0.01161138	0.07285403
20	0.01001580	0.06283823
21	0.00863931	0,05419892
22	0,00745187	0.04674705
23	0.00642755	0.04031950
24	0,00554396	0.03477554
25	0.00478180	0.02999374
26	0.00412437	0,02586937
27	0,00355731	0.02231206
28	0.00306819	0,01924387
29	0,00264632	0,01659755
30	0.00228244	0.01431511
31	0.00196858	0.01234653
32	0.00169788	0.01064864
33	0.00146440	0.00918424
34	0.00126303	0.00792121
35	0.00108934	0,00683187
36	0.00093954	0.00589234
37	0.00081033	0.00508201
38	0.00069890	0.00438311
39	0.00060278	0.00378033
40	0.00051989	0.00326044
41	0.00044839	0.00281205
42	0.00038673	0.00242532
43	0.00033354	0.00209178
44	0.00028767	0.00180411
45	0.00024811	0.00155600
46	0.00021399	0.00134201
47	0.00018456	0.00115745
48	0.00015918	0.00099827
49	0.00013729	0.00086098
50	0.00011841	0.00074257

Seuraava taulukko näyttää samat tiedot, mutta enintään 200 heitolle asti. Todennäköisyydet ovat hyvin pieniä, joten tämä taulukko on tieteellisessä muodossa.

Crapsin selviytymistaulukko 1–200 rullan jaksoille

Rulla Määrä	Todennäköisyys Seitsemän ulos	Todennäköisyys Selviytyminen
1	0.000000E+00	1.000000E+00
2	1.111111E-01	8.888889E-01
3	1.167695E-01	7.721193E-01
4	1.047668E-01	6.673525E-01
5	9.122363E-02	5.761289E-01
6	7.891804E-02	4.972109E-01
7	6.816676E-02	4.290441E-01
8	5.885276E-02	3.701913E-01
9	5.080065E-02	3.193907E-01
10	4.384414E-02	2.755466E-01
11	3.783614E-02	2.377104E-01
12	3.264850E-02	2.050619E-01
13	2.817002E-02	1.768919E-01
14	2.430433E-02	1.525876E-01
15	2.096801E-02	1.316196E-01
16	1.808886E-02	1.135307E-01
17	1.560445E-02	9.792625E-02
18	1.346084E-02	8.446541E-02
19	1.161138E-02	7.285403E-02
20	1.001580E-02	6.283823E-02
21	8.639309E-03	5.419892E-02
22	7.451869E-03	4.674705E-02
23	6.427548E-03	4.031950E-02
24	5.543963E-03	3.477554E-02
25	4.781795E-03	2.999374E-02
26	4.124373E-03	2.586937E-02
27	3.557310E-03	2.231206E-02
28	3.068195E-03	1.924387E-02
29	2.646317E-03	1.659755E-02
30	2.282437E-03	1.431511E-02
31	1.968585E-03	1.234653E-02
32	1.697884E-03	1.064864E-02
33	1.464404E-03	9.184241E-03
34	1.263027E-03	7.921214E-03
35	1.089340E-03	6.831874E-03
36	9.395362E-04	5.892338E-03
37	8.103321E-04	5.082006E-03
38	6.988952E-04	4.383111E-03
39	6.027824E-04	3.780328E-03
40	5.198867E-04	3.260442E-03
41	4.483907E-04	2.812051E-03
42	3.867267E-04	2.425324E-03
43	3.335427E-04	2.091782E-03
44	2.876726E-04	1.804109E-03
45	2.481107E-04	1.555998E-03
46	2.139894E-04	1.342009E-03
47	1.845605E-04	1.157448E-03
48	1.591789E-04	9.982695E-04
49	1.372878E-04	8.609818E-04
50	1.184072E-04	7.425745E-04
51	1.021232E-04	6.404513E-04
52	8.807867E-05	5.523726E-04
53	7.596559E-05	4.764071E-04
54	6.551837E-05	4.108887E-04
55	5.650790E-05	3.543808E-04
56	4.873660E-05	3.056442E-04
57	4.203405E-05	2.636101E-04
58	3.625328E-05	2.273569E-04
59	3.126751E-05	1.960893E-04
60	2.696741E-05	1.691219E-04
61	2.325869E-05	1.458632E-04
62	2.006001E-05	1.258032E-04
63	1.730124E-05	1.085020E-04
64	1.492187E-05	9.358012E-05
65	1.286972E-05	8.071040E-05
66	1.109980E-05	6.961061E-05
67	9.573283E-06	6.003732E-05
68	8.256706E-06	5.178062E-05
69	7.121193E-06	4.465942E-05
70	6.141842E-06	3.851758E-05
71	5.297178E-06	3.322040E-05
72	4.568677E-06	2.865173E-05
73	3.940364E-06	2.471136E-05
74	3.398461E-06	2.131290E-05
75	2.931083E-06	1.838182E-05
76	2.527982E-06	1.585384E-05
77	2.180319E-06	1.367352E-05
78	1.880468E-06	1.179305E-05
79	1.621854E-06	1.017120E-05
80	1.398806E-06	8.772390E-06
81	1.206434E-06	7.565956E-06
82	1.040518E-06	6.525439E-06
83	8.974191E-07	5.628020E-06
84	7.740004E-07	4.854019E-06
85	6.675550E-07	4.186464E-06
86	5.757487E-07	3.610715E-06
87	4.965681E-07	3.114147E-06
88	4.282770E-07	2.685870E-06
89	3.693777E-07	2.316493E-06
90	3.185785E-07	1.997914E-06
91	2.747656E-07	1.723148E-06
92	2.369781E-07	1.486170E-06
93	2.043874E-07	1.281783E-06
94	1.762788E-07	1.105504E-06
95	1.520358E-07	9.534683E-07
96	1.311269E-07	8.223414E-07
97	1.130935E-07	7.092478E-07
98	9.754019E-08	6.117076E-07
99	8.412586E-08	5.275818E-07
100	7.255634E-08	4.550254E-07
101	6.257794E-08	3.924475E-07
102	5.397183E-08	3.384757E-07
103	4.654929E-08	2.919264E-07
104	4.014754E-08	2.517788E-07
105	3.462620E-08	2.171526E-07
106	2.986419E-08	1.872885E-07
107	2.575708E-08	1.615314E-07
108	2.221480E-08	1.393166E-07
109	1.915969E-08	1.201569E-07
110	1.652473E-08	1.036322E-07
111	1.425214E-08	8.938002E-08
112	1.229210E-08	7.708792E-08
113	1.060161E-08	6.648631E-08
114	9.143612E-09	5.734269E-08
115	7.886126E-09	4.945657E-08
116	6.801576E-09	4.265499E-08
117	5.866181E-09	3.678881E-08
118	5.059427E-09	3.172938E-08
119	4.363623E-09	2.736576E-08
120	3.763510E-09	2.360225E-08
121	3.245929E-09	2.035632E-08
122	2.799529E-09	1.755679E-08
123	2.414520E-09	1.514227E-08
124	2.082460E-09	1.305981E-08
125	1.796067E-09	1.126375E-08
126	1.549061E-09	9.714685E-09
127	1.336024E-09	8.378661E-09
128	1.152286E-09	7.226375E-09
129	9.938163E-10	6.232559E-09
130	8.571405E-10	5.375419E-09
131	7.392611E-10	4.636157E-09
132	6.375933E-10	3.998564E-09
133	5.499075E-10	3.448657E-09
134	4.742808E-10	2.974376E-09
135	4.090547E-10	2.565321E-09
136	3.527990E-10	2.212522E-09
137	3.042799E-10	1.908242E-09
138	2.624334E-10	1.645809E-09
139	2.263419E-10	1.419467E-09
140	1.952140E-10	1.224253E-09
141	1.683669E-10	1.055886E-09
142	1.452120E-10	9.106740E-10
143	1.252416E-10	7.854324E-10
144	1.080176E-10	6.774148E-10
145	9.316232E-11	5.842525E-10
146	8.035006E-11	5.039024E-10
147	6.929981E-11	4.346026E-10
148	5.976927E-11	3.748334E-10
149	5.154943E-11	3.232839E-10
150	4.446003E-11	2.788239E-10
151	3.834561E-11	2.404783E-10
152	3.307208E-11	2.074062E-10
153	2.852380E-11	1.788824E-10
154	2.460103E-11	1.542814E-10
155	2.121774E-11	1.330637E-10
156	1.829975E-11	1.147639E-10
157	1.578305E-11	9.898086E-11
158	1.361247E-11	8.536839E-11
159	1.174039E-11	7.362800E-11
160	1.012578E-11	6.350222E-11
161	8.733222E-12	5.476899E-11
162	7.532174E-12	4.723682E-11
163	6.496303E-12	4.074052E-11
164	5.602891E-12	3.513763E-11
165	4.832346E-12	3.030528E-11
166	4.167772E-12	2.613751E-11
167	3.594594E-12	2.254292E-11
168	3.100243E-12	1.944267E-11
169	2.673878E-12	1.676880E-11
170	2.306150E-12	1.446265E-11
171	1.988993E-12	1.247365E-11
172	1.715455E-12	1.075820E-11
173	1.479535E-12	9.278663E-12
174	1.276060E-12	8.002603E-12
175	1.100568E-12	6.902035E-12
176	9.492110E-13	5.952824E-12
177	8.186696E-13	5.134155E-12
178	7.060810E-13	4.428074E-12
179	6.089764E-13	3.819097E-12
180	5.252261E-13	3.293871E-12
181	4.529938E-13	2.840877E-12
182	3.906952E-13	2.450182E-12
183	3.369644E-13	2.113218E-12
184	2.906229E-13	1.822595E-12
185	2.506546E-13	1.571940E-12
186	2.161831E-13	1.355757E-12
187	1.864522E-13	1.169305E-12
188	1.608101E-13	1.008495E-12
189	1.386945E-13	8.698004E-13
190	1.196204E-13	7.501800E-13
191	1.031694E-13	6.470106E-13
192	8.898094E-14	5.580296E-13
193	7.674372E-14	4.812859E-13
194	6.618945E-14	4.150965E-13
195	5.708666E-14	3.580098E-13
196	4.923575E-14	3.087741E-13
197	4.246454E-14	2.663095E-13
198	3.662455E-14	2.296850E-13
199	3.158771E-14	1.980973E-13
200	2.724357E-14	1.708537E-13

Keskimääräinen heittojen määrä ampujaa kohden on 8,525510.

Katso ohjeet tämän tehtävän ratkaisemiseen MathProblems.info -sivustoltani tehtävästä 204.

Kirjoittanut: Michael Shackleford