Texas Hold'em Poker Probability - Paired Flop cont..

From the previous discussion, we assumed that the players play any kind of starting hands.  Or we should say that the probability distribution is the same for each starting hand.

If a player is selective on the starting hands with considering each player's chips, strategy, style, and position,  the probability of each starting hand will be different.

So the estimated probability of the possible 5 card combination (after the flop) is based on the fact (the paired flop) and the probability distribution of the player's starting hand.

For example, if the flop is 99K, and from the past experience player X only plays starting hand 99.  Then we would know the probability of four of a kind is 100% for player X.

Texas Hold'em Poker Probability - Paired Flop cont.

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.

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.