Suited Connector Odds

[DoctorWard]

I want to determine how to calculate the following:
- You hold a suited connector (eg. 8h-9h)

What is the probability of flopping a straight, a flush, a straight flush, a draw with 4-to-a-flush, or a draw with 4-to-a-straight (either open end or inside).

Because it's easier to get a straight or flush with mid suited connectors (4-5 to T-J) you need to weight each hand by 1/13 chance.

Expressed as a percentage and "1 in X". If you can show working it would be helpful.

Any ideas?

[AaronBrown]

There are 50 unknown cards, the flop can be dealt 50*49*48/(3*2*1) = C(50,3) = 19,600 ways.

Let's start with flushes, those are easier. There are 11 cards of your suit. You can deal three of them 11*10*9/(3*2*1) = C(11,3) = 165 ways. So the chance of getting your flush on the flop is 165/19,600 = 0.84% or about 1 in 119.

You can pick two of the 11 cards in 11*10/(2*1) = C(11,2) = 55 ways. If you do that, the third card can be any of the 39 cards from the other suits. 55*39 = 2,145. 2,145/19,600 = 10.94% or about 1 chance in 9.

For straights, if you have AK or A2, there is only one possible straight. KQ and 23 have 2, QJ and 34 have 3, and all others have 4. To get a specific straight on the flop, the first card can be any of 12, the next any of 8 and the last any of 4. 12*8*4/(3*2*1) = 64. So there's 2 chances in 13 of having 64 flops that give you a straight, 2 of having 128, 2 of having 192 and 7 of having 256. On average you have 196.92. 196.92/19,600 = 1.00% or about 1 in 100. But it goes from a low of 0.33% (AK or A2) to a high of 1.31% (anything between 45 and JT).

For a straight flush you have to divide the straight figures by 64.

[DoctorWard]

Aaron, thanks for your help however, I am short a couple of scenarios.

19,600 possible flops with any suited connector.

For 4-5 suited through J-10 suited:
252 combinations give you a straight and only a straight.
161 combinations give you a flush and only a flush.
4 combinations give you a straight flush.

For 3-4 suited and Q-J suited:
189 combinations give you a straight and only a straight.
162 combinations give you a flush and only a flush.
3 combinations give you a straight flush.

For 2-3 suited and K-Q suited:
126 combinations give you a straight and only a straight.
163 combinations give you a flush and only a flush.
2 combinations give you a straight flush.

For A-2 suited and A-K suited:
63 combinations give you a straight and only a straight.
164 combinations give you a flush and only a flush.
1 combination gives you a straight flush.

So as your workings would agree we get to a weighted probability of 1.8308% or 1 in 54.62 of completing a straight, a flush or straight flush.

However, I don't know how to work in the straight draw (open end or inside), 4-to-a-flush draw and straight flush draw. Thoughts?

It seems to get much more complicated!

[BruceZ]

[ QUOTE ]
I want to determine how to calculate the following:
- You hold a suited connector (eg. 8h-9h)

What is the probability of flopping a straight, a flush, a straight flush, a draw with 4-to-a-flush, or a draw with 4-to-a-straight (either open end or inside).

[/ QUOTE ]

This analysis will be for a suited connector that makes the maximum number (4) of straights (54s-JTs). The numbers from the links I've referenced below are known to be accurate, as they have been verified several ways by different people. This earlier analysis has shown that the probability of a flush draw or 8-out straight draw is 19.1%. We need to add the flushes, straights, straight flushes, and gut shots which were not already counted as flush or straight draws.

Straight flushes: 4

Flushes: C(11,3) - 4 = 161

Straights: 4*4^3 - 4 = 252

Flush draws excluding
straights, straight draws, gut shots: 1452

8-out straight draws, including
those that make flush draws or gut shots: 3*(15*34 + 1*27 + 2*6*4) + 4*4*4*2 - 2 = 1881

gut shots excluding
8-out straight draws: 4*(15*30 + 1*24 + 2*6*4) + 2*(15*34 + 1*27 + 2*6*4) = 3258

Explanation of gut shot calculation:

There are 6 pairs of flop cards that give gut shots. For JT, they are AK, AQ, 87, 97, K9, Q8. For AK and 87, it is not possible for the 3rd card to make an open-ended straight draw. For this reason, these two cases are handled in a separate term.

The first term handles the other 4 gut shots. For these, there are 15 ways to pick the gut shot cards without making a flush draw, times 30 cards that do not complete the straight (4), do not pair the board (6), do not make an open-ended draw (4), and do not make a double gut shot (4, they are the 8s for AQ, As for Q8, Ks for 97, and 7s for K9).

Next we consider the 1 combination of suited gut shot cards that make a flush draw times 24 cards that do not complete the straight (4), do not pair the board (6), do not complete an open-ended draw (4), do not make a double gut shot (4, see above), and do not complete a flush (6 since we already subtracted the 1 that makes a straight, 1 that makes and open-ended draw, and 1 that makes a double gut shot).

Next we consider paired boards. There are 2 cards that can pair, 6 ways to make each pair, and 4 ways to pick the unpaired card.

All of this is then repeated for the 2 gut shots with AK and 87, with the only differences being that we no longer need to worry about making an open-ended draw, and instead of subtracting 4 cards for double gut shots, we subtract 4 cards so that we don't count AK9 and Q87 which were already counted by the first term. Note how similar this calculation is to the one above for 8-out straight draws.

Adding all of these flops gives 7008 out of a total of 19600 flops, or 35.76%, or odds of 1.8-to-1.

[BruceZ]

[ QUOTE ]
gut shots excluding
8-out straight draws: 4*(15*30 + 1*24 + 2*6*4) + 2*(15*34 + 1*27 + 2*6*4) = 3258

[/ QUOTE ]

This has been changed in the above post from:

[ QUOTE ]
gut shots excluding
8-out straight draws: 4*(15*34 + 1*27 + 2*6*4) + 2*(15*38 + 1*30 + 2*6*4) - (4*4*4*2 -2) = 3510

[/ QUOTE ]

This was double counting the flops AK9, Q87, AQ8, and K97. The latter two are double gut shots which were subtracted, but only once. The new number agrees with computer simulation done by maddog2030. The explanation has also been modified, and the total has changed from 7260 to 7008.

I have computed the probability of a gut shot draw to be 16.6%. This number differs significantly from the 21.6% in Mike Petriv's book because he a) over counts paired boards, b) counts double gutshots twice instead of 0 times, c) double counts flops (for JTs) AK9 and Q87, d)does not exclude open-ended straight draws, and e) does not exclude flushes.