odds of flopping.. (need a math solution)

[morello]

Hi,

I'm trying to figure out what the odds of flopping any of the following are, given that we are holding 67s (or any suited connector like 89s, JTs)

- two pair(using both hole cards, not a pair on board), trips, full house, quads

- OE straight draw or double belly buster

- flush draw

- made straight or made flush


It's been a while since I've done anything like this, so I'm afraid I wouldn't trust whatever results I did get. Anyone able to do this calculation?

[Cobra]

I will try this but hopefully someone will check my flush draw and open ended straight draws to insure I didn't double count anything. I will use 67 suited.

Straight Flush

There are four different combinations of three cards that give you a straight flush. They are 10,9,8 and 9,8,5 and 8,5,4 and 5,4,3.

So straight flush flops are 4*1*1*1 = 4 flops or .0204%

Four of a kind

= 2*(3c3) = 2 flops or .0102%

Full house with 6 and 7

= 2*(3c1)*(3c1) = 18 flops or .0918%

Flush

= (11c3) = 165 - 4 = 161 flops or .8214% (note you must subtrat out the straight flushes)

Straight

As mentioned before a straight will come with four different combinations of three cards.

= 4*4*4*4 = 256 - 4 = 251 flops or 1.2806%

Three of a Kind

= 2*(3c1)*44 = 264 flops or 1.3469%

Two pair with 6 and 7

= (3c1)*(3c1)*44 = 396 flops or 2.0204%

Flush and an eight out straight draw

The combinations that allow an eight out straight are as follows: 89x, 85x, 54x, 4810, 953. The easiest way to count these combo's is to insure the x doesn't create a straight and it is not the same as one of the other cards. For example:

89x were x is not a 10,5,8 or 9. Now in this combination we need to have two and exactly two of a specific suit. It can be 89 s, x not or 8x s, 9 not or 9x s, 8 not.

89/x = 1*1*27 = 27
8x/9 = 1*7*3 = 21
9x/8 = 1*7*3 = 21
89/9 = 1*1*3 = 3
89/9 = 1*1*3 = 3

Total for 89x is 75 * 3 total combo's = 225

Now for the 10,8,4 and 9,5,3

95/3 = 1*1*3
93/5 = 1*1*3
53/9 = 1*1*3

For a total of 9 flop * 2 combo's = 18

Total of 15 out hands = 225 + 18 = 243 flops or 1.2398%

Flush draw ( 9 outs)

Note some of these have 12 outs to a gutshot straight

= (11c2)*39 = 2145 - 243 = 1902 flops or 9.7041%

Eight out straight draw

We use the same combinations as discussed above but we then subtract out the flops that have a flush already and the flops that have two of our suit. These are already counted.

For 89x

= 4*4*34 = 544 + 2*(4c2)*4 = 544 + 48 = 592

There are three combo's = 1776 flops

for 9,5,3,

= 4*4*4 = 64 * 2 combo's = 128 for a total of 1904 flops

Now there is a total of 26 flops above that give a flush and 243 that have 2 of your suit. Therefor there is
1904-26-243 = 1635 clean flops or 8.3418%

Summary

Straight Flush.......4.......4

Four of a kind.......2.......6.....1 in 3266

Full house..........18......24.....1 in 817

Flush..............161.....185.....1 in 106

Straight...........251.....436.....1 in 45

Three of a Kind....264.....700.....1 in 28

Two Pair w (6,7)...396....1096.....1 in 17.9

15 outs............243....1339.....1 in 14.6

9 outs............1902....3241.....1 in 6.0

8 outs............1635....4876.....1 in 4.0

Cobra

[morello]

Excellent post, thank you!

[man]

In the eight-out straight problem, where did

2*(4c2)*4

come from for 89X? I have only the 544 number as necessary.