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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