With your two hole cards, the other players might have...
Possible starting hands: 1081=47*46/2=C(47,2)
Assuming the flop is 99K,
a) if you flop a three of a kind with 9x
Four of Kind: 0
Three of a Kind or Full House: 1*46+3=49 type 1: 9y, type 2: KK
Two Pair: 3*43+10*6+3=192 type 1: Ky, type 2: yy, type 3: xx
Others: 43*42/2-(10*6+3)=840 any combination without 9 and K without one x card(43*42/2), minus yy and xx pairs.
There is about 22.3% (1 in 4.5) having a two pair or better.
b) if you have a paired starting hand with QQ
Four of Kind: 1
Three of a Kind or Full House: 2*45+3=93 type 1: 9y, type 2: KK
Two Pair: 3*42+10*6+1=187 type 1: Ky (y is any card <> K9), type 2: yy (y<>Q), type 3: QQ
Others: 42*41/2-(10*6+1)=800 any combination without 9s and Ks and two Q cards(42*41/2), minus pairs.
There is about 26% (1 in 4) having a two pair or better. It's slight better than the original 25% that doesn't consider your hole cards.
c) if you flop a two pair with Kx
Four of Kind: 1
Three of a Kind or Full House: 2*45+1=91 type 1: 9y, type 2: KK
Two Pair: 2*43+10*6+3=149 type 1: Ky, type 2: yy ( y not in (9,K,x)), type 3: xx
Others: 43*42/2-(10*6+3)=840
There is about 22.3% (1 in 4.5) having a two pair or better.
d) if your starting hand is XY, X<>Y and not in (9,K)
Four of Kind: 1
Three of a Kind or Full House: 2*45+3=93 type 1: 9y, type 2: KK
Two Pair: 3*42+9*6+2*3=186 type 1: Kz (z is any card not in (9,K)), type 2: zz (z not in (X,Y)), type 3: xx ( x is X or Y)
Others: 42*41/2-(9*6+2*3)=801
There is about 26% (1 in 4.5) having a two pair or better.
9/22/2014
Texas Hold'em Poker Probability - Paired Flop
As we might see a paired flop about 17% of time, let's take a look at what kind of hands players might have.
Possible starting hands: 1176=49*48/2=C(49,2) we can see here the possible starting hands are updated based on the flop (based on the fact).
Four of a Kind: 1
Three of a Kind or Full House: 2*47 + 3 = 97 type 1: pick one card with the same number as the paired card(2) and pick anyone from the remaining card without the same number(47), type 2: pick two cards from with the same number as the unpaired card(3)
Two Pair: 198=3*44+11*6 type 1: pick one with the same number as the unpaired card(3) and pick anyone from the remaining card without the number in the flop(44), type 2: pick any paired card without the number in the flop (11*6)
Others: 880=44*40/2=C(11,2)*4^2 pick any unpaired card without the number in the flop
There is about 25% (1 in 4) having a two pair or better.
* * *
The previous 3 cards or 5 cards analysis give players an idea of what might happen during the game before you see the flop or give a general idea of the distribution of the combinations.
But after you see the flop, the situation is changed. You will have to update the distribution of the combinations based on the flop or based on the fact.
By the way, you also have two cards at hand, so what the other players might have will be different from the previous calculated numbers.
Possible starting hands: 1176=49*48/2=C(49,2) we can see here the possible starting hands are updated based on the flop (based on the fact).
Four of a Kind: 1
Three of a Kind or Full House: 2*47 + 3 = 97 type 1: pick one card with the same number as the paired card(2) and pick anyone from the remaining card without the same number(47), type 2: pick two cards from with the same number as the unpaired card(3)
Two Pair: 198=3*44+11*6 type 1: pick one with the same number as the unpaired card(3) and pick anyone from the remaining card without the number in the flop(44), type 2: pick any paired card without the number in the flop (11*6)
Others: 880=44*40/2=C(11,2)*4^2 pick any unpaired card without the number in the flop
There is about 25% (1 in 4) having a two pair or better.
* * *
The previous 3 cards or 5 cards analysis give players an idea of what might happen during the game before you see the flop or give a general idea of the distribution of the combinations.
But after you see the flop, the situation is changed. You will have to update the distribution of the combinations based on the flop or based on the fact.
By the way, you also have two cards at hand, so what the other players might have will be different from the previous calculated numbers.
9/20/2014
Texas Hold'em Poker Probability - Flop (3 cards)
So what exactly we can see on the flop (3 cards).
There are 22100 combinations, 22100=C(52,3).
Straight Flush: 48=12*4 from A23 to QKA with 4 suits.
Three of a Kind: 52 = 13*4 pick one number from A to K with 4 different suit combinations.
Straight: 720=12*(4^3-4) from A23 to QKA with 4^3 suit combinations minus flush(4).
Flush: 1096=C(13,3)*4-48 pick 3 different numbers from A to K with 4 suits minus straight flush(48).
Pair: 3744=13*6*48 pick one number from A to K with 6 suit combination, pick one from the remaining 48 cards.
Others: 16440= (C(13,3)-12)*(4^3-4) pick 3 different numbers from A to K minus straight, without flush (4^3-4).
So it's about 17% ( 1 in 6 times) to see a pair on the flop.
It's about 26% ( 1in 4 times) to see a pair or better or flush/straight on the flop.
***
We may also want to know what about 2 card flush on the flop, or some kind of straight like patterns on the flop, such as 679, 689, or 579.
2 Card Flush: 12168=C(13,2)*13*3*4 pick two different numbers from A to K for the 2 card flush (C(13,2)), pick another one from A to K for the different suit (13), the suit combination is 4*3.
12168 includes one pair and straight.
So to see a 2 card flush or flush is about 60%, it's quite high, almost more than 1 in 2 games.
What about 2 Card Flush without pair or straight?
It's 9864=(C(13,3)-12)*P(4,2)*3 pick three different numbers from A to K and minus straight(12), the suit combination is P(4,2)*3 (pick two different suits, one for two card flush, one for the other)
***
For straight like combinations, let's consider 3 cards with different suits first
A24~JQA: 264=11*P(4,3) 3 cards all have different suits, P(4,3)
A34~JKA: 264=11*P(4,3)
A24~JQA: 660=11*(P(4,3)+P(4,2)*3) = 11*(4^3-4)
A34~JKA: 660=11*(4^3-4)
There are 22100 combinations, 22100=C(52,3).
Straight Flush: 48=12*4 from A23 to QKA with 4 suits.
Three of a Kind: 52 = 13*4 pick one number from A to K with 4 different suit combinations.
Straight: 720=12*(4^3-4) from A23 to QKA with 4^3 suit combinations minus flush(4).
Flush: 1096=C(13,3)*4-48 pick 3 different numbers from A to K with 4 suits minus straight flush(48).
Pair: 3744=13*6*48 pick one number from A to K with 6 suit combination, pick one from the remaining 48 cards.
Others: 16440= (C(13,3)-12)*(4^3-4) pick 3 different numbers from A to K minus straight, without flush (4^3-4).
So it's about 17% ( 1 in 6 times) to see a pair on the flop.
It's about 26% ( 1in 4 times) to see a pair or better or flush/straight on the flop.
***
We may also want to know what about 2 card flush on the flop, or some kind of straight like patterns on the flop, such as 679, 689, or 579.
2 Card Flush: 12168=C(13,2)*13*3*4 pick two different numbers from A to K for the 2 card flush (C(13,2)), pick another one from A to K for the different suit (13), the suit combination is 4*3.
12168 includes one pair and straight.
So to see a 2 card flush or flush is about 60%, it's quite high, almost more than 1 in 2 games.
What about 2 Card Flush without pair or straight?
It's 9864=(C(13,3)-12)*P(4,2)*3 pick three different numbers from A to K and minus straight(12), the suit combination is P(4,2)*3 (pick two different suits, one for two card flush, one for the other)
***
For straight like combinations, let's consider 3 cards with different suits first
A24~JQA: 264=11*P(4,3) 3 cards all have different suits, P(4,3)
A34~JKA: 264=11*P(4,3)
A35~TQA: 240=10*P(4,3)
If we include 2 card flush but not 3 card flush,
A24~JQA: 660=11*(P(4,3)+P(4,2)*3) = 11*(4^3-4)
A34~JKA: 660=11*(4^3-4)
A35~TQA: 600=10*(4^3-4)
So, we might see straight or straight-like flops about 12% ( 1 in 8 times).
So, we might see straight or straight-like flops about 12% ( 1 in 8 times).
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