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.

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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)

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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).