Quote:
Originally Posted by OMGClayDol
what about say 97s 86s etc?
I think you are asking what is the probability of making a straight flush in hold-em with a suited one-gapper where you must use one or both of your hole cards.
It might be easiest to simply walk through every case. Of course, there are C(50,5) = 2,118,760 possible 5-card boards for any 2-card hole card combination, so that is the denominator in the probability. What follows are the numerators.
AQ
A-T: KJT on board -> C(3,3)*C(47,2) = 1081
Q-8: JT98 on board, not K -> C(4,4)*C(45,1) = 45
5-A: 5432 on board, not 6 -> C(4,4)*C(45,1) = 45
Total = 1171
KJ
A-T: AQT on board -> C(3,3)*C(47,2) = 1081
K-9: QT9 on board, not A -> C(3,3)*C(46,2) = 1035
J-7: T987 on board, not Q -> C(4,4)*C(45,1) = 45
Total = 2161
QT
A-T: AKJ on board -> C(3,3)*C(47,2) = 1081
K-9: KJ9 on board, not A -> C(3,3)*C(46,2) = 1035
Q-8: J98 on board, not K -> C(3,3)*C(46,2) = 1035
T-6: 9876 on board, not J -> C(4,4)*C(45,1) = 45
Total = 3196
J9
A-T: AKQT on board -> C(4,4)*C(46,1) = 46
K-9: KQT on board, not A -> C(3,3)*C(46,2) = 1035
Q-8: QT8 on board, not K -> C(3,3)*C(46,2) = 1035
J-7: T87 on board, not Q -> C(3,3)*C(46,2) = 1035
9-5: 8765 on board, not T -> C(4,4)*C(45,1) = 45
Total = 3196
T8
A-T: AKQJ on board -> C(4,4)*C(46,1) = 46
K-9: KQJ9 on board, not A -> C(4,4)*C(45,1) = 45
Q-8: QJ9 on board, not K -> C(3,3)*C(46,2) = 1035
J-7: J97 on board, not Q -> C(3,3)*C(46,2) = 1035
T-6: 976 on board, not J -> C(3,3)*C(46,2) = 1035
8-4: 7654 on board, not 9 -> C(4,4)*C(45,1) = 45
Total = 3241
97
K-9: KQJT on board, not A -> C(4,4)*C(45,1) = 45
Q-8: QJT8 on board, not K -> C(4,4)*C(45,1) = 45
J-7: JT8 on board, not Q -> C(3,3)*C(46,2) = 1035
T-6: T86 on board, not J -> C(3,3)*C(46,2) = 1035
9-5: 865 on board, not T -> C(3,3)*C(46,2) = 1035
7-3: 6543 on board, not 8 -> C(4,4)*C(45,1) = 45
Total = 3240
86
Q-8: QJT9 on board, not K -> C(4,4)*C(45,1) = 45
J-7: JT97 on board, not Q -> C(4,4)*C(45,1) = 45
T-6: T97 on board, not J -> C(3,3)*C(46,2) = 1035
9-5: 975 on board, not T -> C(3,3)*C(46,2) = 1035
8-4: 754 on board, not 9 -> C(3,3)*C(46,2) = 1035
6-2: 5432 on board, not 7 -> C(4,4)*C(45,1) = 45
Total = 3240
75
J-7: JT98 on board, not Q -> C(4,4)*C(45,1) = 45
T-6: T986 on board, not J -> C(4,4)*C(45,1) = 45
9-5: 986 on board, not T -> C(3,3)*C(46,2) = 1035
8-4: 864 on board, not 9 -> C(3,3)*C(46,2) = 1035
7-3: 643 on board, not 8 -> C(3,3)*C(46,2) = 1035
5-A: 432A on board, not 6 -> C(4,4)*C(45,1) = 45
Total = 3240
64
T-6: T987 on board, not J -> C(4,4)*C(45,1) = 45
9-5: 9875 on board, not T -> C(4,4)*C(45,1) = 45
8-4: 875 on board, not 9 -> C(3,3)*C(46,2) = 1035
7-3: 753 on board, not 8 -> C(3,3)*C(46,2) = 1035
6-2: 532 on board, not 7 -> C(3,3)*C(46,2) = 1035
Total = 3195
53
9-5: 9876 on board, not T -> C(4,4)*C(45,1) = 45
8-4: 8764 on board, not 9 -> C(4,4)*C(45,1) = 45
7-3: 764 on board, not 8 -> C(3,3)*C(46,2) = 1035
6-2: 642 on board, not 7 -> C(3,3)*C(46,2) = 1035
5-A: 42A on board, not 6 -> C(3,3)*C(46,2) = 1035
Total = 3195
42
8-4: 8765 on board, not 9 -> C(4,4)*C(45,1) = 45
7-3: 7653 on board, not 8 -> C(4,4)*C(45,1) = 45
6-2: 653 on board, not 7 -> C(3,3)*C(46,2) = 1035
5-A: 53A on board, not 6 -> C(3,3)*C(46,2) = 1035
Total = 2160
A3
A-T: KQJT on board -> C(4,4)*C(46,1) = 46
7-3: 7654 on board, not 8 -> C(4,4)*C(45,1) = 45
6-2: 6542 on board, not 7 -> C(4,4)*C(45,1) = 45
5-A: 542 on board, not 6 -> C(3,3)*C(46,2) = 1035
Total = 1171
I think these are correct as I tried to be both methodical and careful. But if anybody sees anything that looks like it could be wrong, don't hesitate to say so.