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flush or str8 draw on suited connectors, Aisthesis, 11. Nov 2003 10:34
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I took some shortcuts in calculating this but wondered if the result is close to correct. The question is: With (middle!) suited connectors, what are the probabilities of flopping an open-ended straight draw or a 4-flush?
1) Open-ended straight. There are (50)(49)(48)/6 possible flops (excluding permutations). Taking 98 as example, there should be (4)(4)(48) ways of getting JT (I know I'm counting a few hands more than once but hope it's not so severe as to throw the result off), same for T7 and 76. This works out to just a little over 10%. Does that sound about right?
2) Flush draw, including actual flush. Same number of possible flops. Ordering the hands according to the desired suit, I get (11)(10)(48)/2 possible 4- or 5-flushes, which works out to over 13%. How far off am I there?
That should mean that there's a chance of over 20% that you'll get one or the other--and then somewhere around 35% after that that you'll make your draw by the river.
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Re: flush or str8 draw on suited connectors, pt_Gatsby, 12. Nov 2003 10:54
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This is actually a hard question. The problem is with the straight draw. Assuming middle pairs (no A/K, no 3/2), there are quite a few ways where you can achieve an open straight.

But, to extend the logic.

There are 50_c_3 flops available, or 19,600.


There are now 3 seperate ways to get the straight: Two above, Two below, one above and one below. And this is where my head starts to hurt.


So the chance of getting the 1st card is: a1) Card touching either side of you (8/50), b1) card seperated by one (8/50) c1) dead card (34/50).

So, for each of these:

Get a touching card, Card 2 and 3 will result in:

a1.2) Card above or below: 8/49 (complete: 84/2450)
a1.2) Dead card: 42/49 (incomplete: 336/2450)

a1.3)(currently at 336/2450) Above or below: 8/48 (2688/117600)
a1.3) 40/48 (13440/117600) Dead card.

Assuming touching first card, then;
84/2450 + 2688/117600 = get a straight
336/2450 + 13400/117600 = not get a straight


Get a seperated card, Possibilities are: Get the one missing card in the next 2 cards, OR get two lower cards. This one is easier, surprisingly.

b1.2) 4/49 chance of getting the card needed. (8/50*4/49 = 32/2450)
b1.2a) 4/49 chance of getting a split lower cards (8/50*4/49= 32/2450)
b1.2) 41/49 chance of not getting the card needed (not including those included next)

b1.3) 4/48 chance of getting the card needed when blank. (8/50*41/49*4/48 = 1312/117600)
b1.3a) 4/48 chance of getting card when a split card was hit. (8/50*4/49*4/48 = 128/117600)
b1.3) 40/48 chance of not getting a card needed (8/50* 41/49*40/48 = 13120/117600

Assuming Split first card, then;
(32/2450 + 1312/117600) + (32/2450 + 128/117600) = get a straight
336/2450 + 13400/117600 = not get a straight

And finally, the dead card first:

Need two exact cards in many combinations to hit. They are: 2 above, 2 below and 2 split, which results in 8/49*4/48 * 3.


Assuming dead first card, then:
(34*50) * 96/2352 = Get a straight
(34*50) * 2256/2352 = Not a straight

Sooooo....

The total combinations are:

(84/2450 + 2688/117600) + [(32/2450 + 1312/117600) + (32/2450 + 128/117600)] + [34/50 * 96/2352]

To verify this:

The logic goes:

34/50 times - you get a dead card. Check - only 16 card help you.
8/50 times - you get a card that touches. Check - only 8 cards that touch either side
8/50 times - you get a card that is seperated by one. Check, only 8 cards can be seperated by one

Of the 34/50 times you miss on the first card, you may get the last two cards to make your straight

Of the 8/50 times the first card touches, you now have 8/49 cards that will make it.

Of the 8/50 times the first card is split, you will have 4/49 cards that will make it.
Of the 8/50 times the first card is split, you have a chance to get two lower cards to make your straight

Of the times the first card touches, and the second fails, you have a 8/48 that the third will make it.

Of the time the first card is split, and the second fails, you have a 4/48 that the third will make it.

I think that's right:

That leaves you a 12.3265% chance of getting a 4 running straight. I think. Sounds correct, however.


---

As for the flush, I'm gonna do it shorthand, heh. There are 11 remaining suited cards. Your chances are:

11/50 * 10/49 * 9/48 - autoflush
11/50 * 10/49 * 1 - runner flush (combination 1)
11/50 * 10/49 * 1 - runner flush (combination 2)
11/50 * 10/49 * 1 - runner flush (combination 3)


Or: 4.489795918 * 3 chances of getting a runner flush, and 0.84% chance of tetting a flush.
So, I can conclude roughly this:

There is a 12.3% chance of getting a 4 straight,including the possibility of a full straight.
There is a 13.1% chance of getting a 4 flush, including the possibility of a full flush, including in this set, the possibility of getting the straight.



Whoa. That might... maybe... perhaps... even be right. It sound approximately right, considering how often a straight and flush hand is made from a suited connector. However, no warranty in this one!


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Re: flush or str8 draw on suited connectors, Aisthesis, 12. Nov 2003 21:25
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Yes, it is hard. Glad your results were very close to mine, although your 12% on the straight draw would mean I was 2% too low (I think I may have deducted a little for some reason).
While it creates certain calculation risks, I think it makes it easier if you count ORDERED flops (excluding permutations). That was the reason I divided by 6 (=3!) in counting possible flops. Then you really just have 3 cases to deal with: 2 connectors above, 1 card on each end, and 2 connectors below. I think that was the reason I deducted actually, because I counted some of these hands twice but was too lazy to figure out exactly how many.
For me, anyway, I'm happy to know (we both ended up with this much) that, given a middle suited connector, the odds of flopping an open straight draw (including the actual straight, which will be rare) or a 4-flush (including the flush, also rare) are better than 1/5 and worse than 1/4.
Also, for GETTING the straight on turn or river, aren't the odds something like 31%? For the flush I'm sure they're right at 35%.
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Re: flush or str8 draw on suited connectors, Schuster, 13. Nov 2003 05:08
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Note that the combinations that result in a double belly buster draw... 4^3 for each one, and there's 2 possibilities for each, so that adds 128 (I think that's right) combos. Adds less than 1% but it helps.

Lee
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Re: flush or str8 draw on suited connectors, Aisthesis, 13. Nov 2003 11:25
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Ok, I didn't entirely follow your logic on the straight, but I think we're good on the 4-flush: 13% chance or slightly better than 1:7 odds.
Let me try for a truly precise calculation on the open-ended straight give a 98 pocket:
(50)(49)(48)/6 possible flops (let's label this constant "f")
1) Case of JT: I'm going to order these with a J or a T first and exclude J's and T's on card 3: That gives (4)(4)(42) possibilities. Plus we have JJT and TTJ flops: There are (4)(3)(4)/2 of each. So let's create a constant s=(4)(4)(42)+(4)(3)(4). That's the number of open-ended straights (and straights) with high cards.
2) s will also be the number of open-ended straights in the other two cases, BUT:
3) We've counted all the flops that simultaneously count in two different cases twice (impossible for any flop to count in all three). These are as I see it exactly the made straights, of which there are (4)(4)(4)(2).
Result is a probability of [3s - (4)(4)(4)(2)]/f =5/49 or 10.2%
I think that should be right on the money.
So the ODDS (thanks, Lee, for improving my way of thinking about this!) are about 9:1
Adding the odds of straight draw and flush draw we get 23%, but we have to deduct 1% for cases where both occur (I know this will work as rough estimate, and in fact I think it's actually completely correct since rank and suit should be completely independent variables). So the chances of getting one draw or the other are 22%. Odds, then, are surprizingly favorable (to me anyway) at 3.5:1
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str8 draws with 1- and 2-gappers, Aisthesis, 15. Nov 2003 21:58
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Here, I'm talking about flopping an open straight draw or straight with a 97 or a 96. Just as a note, the probability of an open straight draw with 43 should be the same as with 97, since the downward limitation gives exactly the same number of card combination as far as I can tell.
First, an estimate, which I'm guessing is right to within 0.1% (I'll check it in a separate post): I figured 10.2% for getting such a flop with a 98, so with 97, looking at the previous calculation, we are basically just taking out one of the 3 straight combinations; hence, the probability should be 2/3 of 10.2% or 6.8%
With a 2 gapper, we're down to just one combo (for open end straight, we MUST have both a 7 and an 8 on the flop with a 96), so it should be 3.4%
I think this is an argument for being a lot more restrictive with 96s than with 97s or 98s.
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Re: flush or str8 draw on suited connectors, Aisthesis, 15. Nov 2003 22:13
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1) One-gappers using constants from my last post on immediate connectors. The probability should be [2s - (4)(4)(4)]/f = 7.0% So, my rough estimate was 0.2% too low--more than I thought, actually.
2) Two-gappers: s/f = 9/245 or 3.7%, so my estimate was too low by 0.3% here.
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Re: flush or str8 draw on suited connectors, Aisthesis, 16. Nov 2003 17:46
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If we include the double belly-buster flops here, I get (for connectors like 98): 27/245=11.0%
And for one-gappers (97, etc.): 18/245=7.3%
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Re: flush or str8 draw on suited connectors, CRCarson, 29. Nov 2003 05:04
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Impressive piece of work that. Is there a book that has this kind of stuff exhaustively detailed?
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Re: flush or str8 draw on suited connectors, Aisthesis, 29. Nov 2003 11:56
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Hey, thanks! There probably is, but I don't know of it. There are a number of useful odds calculations done by Mike Caro in the back of SuperSystem, and I think you can call up some of the same tables with the links here. I think there was a small fudge factor on my flush calculation, and certainly on combining the flush OR straight draw calculation, but I'm pretty certain it's small enough not to matter in practice (within 0.2%). The straight draw part really should be right on the money.
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