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Texas
Holdem Post Flop Odds Math
QQ Not Having A or K Hit by the River
Hypothesis: own hand dealt QQ. Event: A or K not hit by the river
First, the probability for A or K to be hit by the river:
Favorable combinations for this event (final board configuration):
(Axyzt), in number of 4*C(42,4)
(AAxyz), in number of 6*C(42,3)
(AAAxy), in number of 4*C(42,2)
(AAAAx), in number of 1*42
(Kxyzt), in number of 4*C(42,4)
(KKxyz), in number of 6*C(42,3)
(KKKxy), in number of 4*C(42,2)
(KKKKx), in number of 1*42
(AKxyz), in number of 16*C(42,3)
(AKKxy), in number of 4*6*C(42,2)
(AKKKx), in number of 4*4*42
(AKKKK), in number of 4
(AAKxy), in number of 6*4*C(42,2)
(AAKKx), in number of 6*6*42
(AAKKK), in number of 6*4
(AAAKx), in number of 4*4*42
(AAAKK), in number of 4*6
(AAAAK), in number of 4
(where x, y, z and are different from A and K)
Totally, we have 1268092 favorable combinations for the event “A or K hit by river”.
The total number of possible combinations for the final board: C(50,5) = 2118760.
The probability: 1268092/2118760 = 59.850%
Then, the probability for the event “A or K not hit by the river” is 1 - 1268092/2118760 =
40.149% = 1 / 2.490
QQ versus AK Heads Up, A or K Hitting by River
Hypothesis: own hand dealt QQ, one opponent dealt AK. Event: A or K hit by river.
We have 3 aces and 3 kings left.
The favorable combinations of the final board for the event to measure:
(Axyzt), in number of 3*C(42,4)
(AAxyz), in number of 3*C(42,3)
(AAAxy), in number of C(42,2)
(Kxyzt), in number of 3*C(42,4)
(KKxyz), in number of 3*C(42,3)
(KKKxy), in number of C(42,2)
(AKxyz), in number of 9*C(42,3)
(AKKxy), in number of 3*3*C(42,2)
(AKKKx), in number of 3*42
(AAKxy), in number of 3*3*C(42,2)
(AAKKx), in number of 3*3*42
(AAKKK), in number of 3*1
(AAAKx), in number of 3*42
(AAAKK), in number of 1*3
(x, y, z and t are different from A and K).
Totally, we have 861639 favorable combinations, from C(48,5) = 1712304 possible.
Probability: 861639/1712304 = 50.520% = 1 / 1.987
Flop Being All on Kind
Hypothesis: no cards seen.
For each card (as value), we have C(4,3)=4 favorable combinations for the event. Totally, 13*4 = 52 favorable combinations, from C(52,3) = 22100 possible.
Probability: 52/C(52,3) = 0.235% = 1 / 425
Four Flush Improving
Hypothesis: 4 same symbols in own hand plus flop. Event: at least one symbol hit by river. Let S be the same symbol of that four cards.
Favorable 2-card combinations of turn plus river:
(Sx), with x different from S, in number of 9*(47-9) and
(SS), in number of C(9,2) = 9*4.
Totally, we have 378 favorable combinations from C(47,2) = 1081 possible.
Probability: 378/1081 = 34.967% = 1 / 2.859
Open ended straight flush improving
Hypothesis: 4 cards from a straight flush (not starting with 2 or 10) in own hand plus flop. Event: one or two of the 2 cards left to be hit by river.
Let C and D be the needed cards (unique cards – value + symbol).
Favorable combinations of turn and river:
(Cx), with x different from C and D, in number of 47-2
(Dx), with x different from C and D, in number of 47-2
(CD), in number of 1
Totally, we have 91 favorable combinations from C(47,2) = 1081 possible.
Probability: 91/1081 = 8.418% = 1 / 11.879
Open Ended Straight Improving
Hypothesis: 4 cards from a flush (not starting with 2 or 10) in own hand plus flop. Event: one or two of the 2 cards left (as value i.e. 8 outs) to be hit by river.
Let C and D be the needed cards (as value).
Favorable combinations of turn and river:
(Cx), with x different from C and D, in number of 4*(47-8)
(Dx), with x different from C and D, in number of 4*(47-8)
(CD), in number of 8
(CC), in number of C(4,2)
(DD), in number of C(4,2).
Totally, we have 332 favorable combinations from C(47,2) = 1081 possible.
Probability: 332/1081 = 30.712% = 1 / 3.256 .
Two pair making a full house
Hypothesis: two pair in own hand plus flop. Event: Making a full house by river
Let’s denote by (TTDDC) the cards from own hand and flop, T, D and C mutually different (as value).
Favorable combinations of turn plus river for a full house to make:
(Tx), x different from T, D and C, in number of 2*(47-2-2-3)
(Dx), x different from T, D and C, in number of 2*(47-2-2-3)
(TT), in number of 1
(TD), in number of 2*2
(DD), in number of 1
(CC), in number of C(3,2)
(TC), in number of 2*3
(DC), in number of 2*3.
Totally, we have 181 favorable combinations from C(47,2) = 1081 possible.
Probability: 181/1081 = 16.743% = 1 / 5.972
Trips Improving to Full House or Better
Hypothesis: three same cards (as value) in own hand plus flop Event: making a full house or quads by river.
Let’s denote by (TTTCD) the cards (T, C and D mutually different – as value).
Favorable combinations of turn plus river:
(Cx), x different from T, C and D, in number of 3*(47-1-3-3)
(Dx), x different from T, C and D, in number of 3*(47-1-3-3)
(CT), in number of 3*1
(DT), in number of 3*1
(DC), in number of 3*3
(xx), x different from T, C and D, in number of 10*C(4,2)
(CC), in number of C(3,2)
(DD), in number of C(3,2)
(Tx), x different from T, C and D, in number of 47-1-3-3.
Totally, we have 361 favorable combinations, from C(47,2) = 1081 possible.
Probability: 361/1081 = 33.395% = 1 / 2.994 .
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