[ QUOTE ]
I am wondering the odds that given two players each hold a different pocket pair they split the pot with 4-of-a-kind.
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It depends on the rank of the higher pocket pair. The higher it is, the less likely the pot will be split, since splitting requires that the kicker be greater than or equal to the higher pocket pair.
If the higher pair is a pair of N's (J = 11, Q = 12, K = 13, A = 14) then the probability of a quad split is:
{ [(14-N)*4 + 2]*(N-3) + [(13-N)*4 + 2]*(14-N) } / C(48,5)
That is, for each of the N-3 possible quads less than N, there are (14-N)*4 kickers greater than N, plus 2 of rank N. Then for each of the 14-N quads greater than N, there are (13-N)*4 kickers greater than N, plus 2 of rank N.
<font class="small">Code:</font><hr /><pre>
high pair P(split) odds-to-1
33 0.027% 3705
44 0.025% 4057
55 0.022% 4481
66 0.020% 5006
77 0.018% 5669
88 0.015% 6535
99 0.013% 7712
TT 0.011% 9407
JJ 0.0083% 12057
QQ 0.0060% 16786
KK 0.0036% 27617
AA 0.0013% 77831
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