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how many tournaments do you have to play before you get heads up with any other given player

The number of ways to choose r elements from a set of n elements is given by the Binomial Coefficient formula:

n!/(r!(n-r)!)

So, for example, if we have a tournament with 10 players starting, the calculation for the number of possible pairs of players that could get heads up is:

10!/(2!8!)
3628800/(2 * 40320)
3628800/80640
45

So there are 45 possible pairs of players. If we assume that each player has an equal probability of being first, and each player has an equal probability of being second then we can calculate the probably of any given pair of players getting heads up.

p = 1/45 = 0.0222

We can also calculate how many tournaments on average we would expect to have to play before a given pair gets heads up:

1/p = 45

The probability of any 2 given players not getting heads up in a tournament is

1-(1/45) = 44/45

The probability of 2 players having not got heads up after t tournaments is:

(1-p)^t

Probability table for the probability of not having got heads up after a given number of tournaments (assuming random distribution):

Trials  Prob Not HU     Prob HU 
1	0.9777777778	0.0222222222
2	0.9560493827	0.0439506173
3	0.9348038409	0.0651961591
4	0.9140304222	0.0859695778
5	0.893718635	0.106281365
6	0.8738582209	0.1261417791
7	0.8544391493	0.1455608507
8	0.8354516127	0.1645483873
9	0.8168860213	0.1831139787
10	0.7987329986	0.2012670014
11	0.7809833764	0.2190166236
12	0.7636281903	0.2363718097
13	0.7466586749	0.2533413251
14	0.7300662599	0.2699337401
15	0.7138425653	0.2861574347
16	0.6979793971	0.3020206029
17	0.6824687439	0.3175312561
18	0.6673027718	0.3326972282
19	0.6524738213	0.3475261787
20	0.6379744031	0.3620255969
21	0.6237971941	0.3762028059
22	0.6099350342	0.3900649658
23	0.5963809224	0.4036190776
24	0.583128013	0.416871987
25	0.5701696127	0.4298303873
26	0.5574991768	0.4425008232
27	0.5451103063	0.4548896937
28	0.5329967439	0.4670032561
29	0.5211523718	0.4788476282
30	0.509571208	0.490428792
31	0.4982474034	0.5017525966
32	0.4871752388	0.5128247612
33	0.4763491224	0.5236508776
34	0.4657635864	0.5342364136
35	0.4554132845	0.5445867155
36	0.4452929892	0.5547070108
37	0.4353975895	0.5646024105
38	0.4257220875	0.5742779125
39	0.4162615967	0.5837384033
40	0.407011339	0.592988661
41	0.3979666425	0.6020333575
42	0.3891229394	0.6108770606
43	0.3804757629	0.6195242371
44	0.372020746	0.627979254
45	0.3637536183	0.6362463817
46	0.3556702046	0.6443297954
47	0.3477664222	0.6522335778
48	0.3400382795	0.6599617205
49	0.3324818733	0.6675181267
50	0.3250933872	0.6749066128
51	0.3178690897	0.6821309103
52	0.3108053322	0.6891946678
53	0.303898547	0.696101453
54	0.297145246	0.702854754
55	0.2905420183	0.7094579817
56	0.284085529	0.715914471
57	0.2777725172	0.7222274828
58	0.2715997946	0.7284002054
59	0.2655642436	0.7344357564
60	0.259662816	0.740337184
61	0.2538925312	0.7461074688
62	0.248250475	0.751749525
63	0.2427337977	0.7572662023
64	0.2373397133	0.7626602867
65	0.2320654975	0.7679345025
66	0.2269084864	0.7730915136
67	0.2218660756	0.7781339244
68	0.2169357184	0.7830642816
69	0.2121149246	0.7878850754
70	0.2074012597	0.7925987403
71	0.2027923428	0.7972076572
72	0.1982858463	0.8017141537
73	0.1938794941	0.8061205059
74	0.1895710609	0.8104289391
75	0.1853583707	0.8146416293
76	0.1812392958	0.8187607042
77	0.1772117559	0.8227882441
78	0.1732737169	0.8267262831
79	0.1694231898	0.8305768102
80	0.16565823	0.83434177
81	0.161976936	0.838023064
82	0.1583774486	0.8416225514
83	0.1548579497	0.8451420503
84	0.1514166619	0.8485833381
85	0.1480518472	0.8519481528
86	0.1447618062	0.8552381938
87	0.1415448772	0.8584551228
88	0.1383994354	0.8616005646
89	0.1353238924	0.8646761076
90	0.1323166948	0.8676833052
91	0.1293763238	0.8706236762
92	0.1265012944	0.8734987056
93	0.1236901545	0.8763098455
94	0.1209414844	0.8790585156
95	0.1182538959	0.8817461041
96	0.1156260315	0.8843739685
97	0.1130565642	0.8869434358
98	0.1105441961	0.8894558039
99	0.1080876584	0.8919123416
100	0.1056857104	0.8943142896
101	0.1033371391	0.8966628609
102	0.1010407582	0.8989592418
103	0.098795408	0.901204592
104	0.0965999545	0.9034000455
105	0.0944532889	0.9055467111
106	0.0923543269	0.9076456731
107	0.0903020085	0.9096979915
108	0.0882952972	0.9117047028
109	0.0863331795	0.9136668205
110	0.0844146644	0.9155853356
111	0.082538783	0.917461217
112	0.0807045878	0.9192954122
113	0.0789111525	0.9210888475
114	0.0771575713	0.9228424287
115	0.0754429586	0.9245570414
116	0.0737664484	0.9262335516
117	0.072127194	0.927872806
118	0.0705243675	0.9294756325
119	0.0689571593	0.9310428407
120	0.067424778	0.932575222
121	0.0659264496	0.9340735504
122	0.0644614174	0.9355385826
123	0.0630289415	0.9369710585
124	0.0616282983	0.9383717017
125	0.0602587806	0.9397412194
126	0.0589196966	0.9410803034
127	0.05761037	0.94238963
128	0.0563301395	0.9436698605
129	0.0550783587	0.9449216413
130	0.0538543951	0.9461456049
131	0.0526576308	0.9473423692
132	0.0514874612	0.9485125388
133	0.0503432954	0.9496567046
134	0.0492245555	0.9507754445
135	0.0481306765	0.9518693235
136	0.0470611059	0.9529388941
137	0.0460153036	0.9539846964
138	0.0449927413	0.9550072587
139	0.0439929026	0.9560070974
140	0.0430152825	0.9569847175
141	0.0420593873	0.9579406127
142	0.0411247343	0.9588752657
143	0.0402108513	0.9597891487
144	0.0393172768	0.9606827232
145	0.0384435596	0.9615564404
146	0.0375892582	0.9624107418
147	0.0367539414	0.9632460586
148	0.0359371871	0.9640628129
149	0.035138583	0.964861417
150	0.0343577256	0.9656422744

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