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Apparently the odds of flopping a flush draw are 10.94% for suited connectors and 10.45% for hitting an 8-out (open-ended, I don't know about double gut-shot) straight draw. Could someone point to a resource or do it by themselves where they explain the odds of hitting either a flush draw or a straight draw or both. You can't simply add the two probabilities. There's some but not total overlap between them so it should be somewhere between 11% and 22%, but I can't find an answer. Also what would happen if you include gut-shots.
You could think about how many different combinations you can flop a straight draw with at most one card of the three having a club if your suited connector hole cards are clubs for example. Then divide that number of combinations by the total number of flop combinations, which is 19,600. Add that to the flush draw percentage (now there would be no over-lap) of 10.94%. If anyone is familiar with combinatorial mathematics they can check out my calculation of 8.54% for the probability of a non=flush straight draw. If any flaws can be pointed out with the calculation that would be helpful A link to my own mathematical calculations (the word download link is on the right), The reason I'm asking this though is I'm not sure the number of combinations calculated has no repeated occurrences; in other words it might be bigger than the actual number of combinations. Someone linked to a post that had 7.6% so maybe there's something off in the math, or if someone else posts a mathematical answer that seems definitively correct, than I don't need these calculations anymore

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  • here's a good resource
    – user1165
    Jul 2, 2015 at 16:32
  • hmm, i calculated it myself and got 8.4% instead of 7.8%
    – user3163829
    Jul 2, 2015 at 18:08
  • are you referring to this formula 3*(4*4*34 – 1*1*7 -1*1*27 -2*7*3 +2*(4c2*4-3))/50c3 ~ 7.80% ? That's only for suited 0-gappers suited, which is what you want as 0-gapper = suited connectors. I have to say i'm not fond of these formulas to give an answer but i'm quite eager to run a simulation in my evaluator to find an answer, although that won't have any mathematical explanation
    – user1165
    Jul 2, 2015 at 19:32
  • i'd be interested in the results of a simulation, and which percentage it lied closer to
    – user3163829
    Jul 3, 2015 at 1:33
  • pocketfives.com/f7/… Someone has also run a large sample simulation here. It looks like there is only a 1.02% chance of flopping an 8 out straight draw with a flush draw with connectors. The odds of either a flush draw or a straight draw would then be: 10.94+10.42-1.02=20.34% or 22.48% with made flushes and straights included. Would be interested to still see a mathematical answer -beyond my ability though
    – Chris
    Jul 3, 2015 at 11:29

5 Answers 5

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Well you can add if the flush was excluded from the straight odds calculation. Just as straight-flush should be removed from the straight calculation as it is s different hand.

This is a flush draw using combination
You have 50 cards left and from a set of 3 you want two that are the same suite
The suite you want has 11 cards
39 is the count of other cards = 50-11
(11/2) is not a fraction - it is binomial coefficient
4 flush (11/2)(39/1)/(50/3) = 0.1094 = 8.1 : 1

If you add in flopping the whole flush (11/3) then you get the 0.11765 = 7.5 : 1 you are more used to seeing
4 or five flush ((11/2)(39/1) + (11/3) - 4) / (50/3)
the - 4 is straight flush
yyy xx
yy xx y
y xx yy
xx yyy

Calculating a straight draw is much more difficult

As for 4 draw you would just add them as a straight flush can still make a straight and not a flush and visa versa
Now as for a the 5 card hand you would need to discount out the straight-flush from either as it is a separate hand

straight or 8 outer straight draw
The straight answer is 0.1171 = 7.54 : 1

And you can just add as the straight calc above excludes flush
0.1171 + 0.11765 = 0.2344 = 3.27 : 1

Your odds of getting suited connector is 24.5 : 1
If you draw a suited connector and can get in for the right price then it is worth seeing a flop

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Another attempt at this. I think I finally understand the question with a little help from comments from Chris :)

So you are looking for the odds in the situation where you have 2 suited hole cards pre-flop and the flop comes with two suited cards to match yours and one that does not, giving you 4 of the same suit and a flush draw.

Before the flop comes out there are 11 cards that can help your flush out of a total of 50 remaining cards, so the odds of the first card helping your flush are 11/50 or 22%.

On the second flopped card, we will assume the first hit and this one will miss. Reducing the remaining cards by 1. Leaving 48 possible cards and 10 of those are going to help you.

So on the third card, the odds of hitting are 10/48 or about 21%.

In order to calculate the odds of both of those hitting, you would use

11/50 * 10/48 = 4.6%

I'll do the odds for the other possible combinations...

1st and 2nd hit = 11/50 * 10/49 = 4.5% 1st and 3rd hit = 11/50 * 10/48 = 4.6% 2nd and 3rd hit = 11/49 * 10/48 = 4.7%

As you can see, as the pool of remaining cards gets smaller without hitting your card, the odds of hitting it go up on the next card.

Also, worthy of note I think, I was playing on Carbon Poker the other day and they have a free odds calculator program for use on their site. This shows you the odds of hitting different hands based on your hole cards. Has cool little bar graphs and everything. You might want to check that out (they have play money tables, but not sure if the odds calculator works on those or not. I'm assuming it does). Even if it doesn't show you the odds of getting a flush DRAW, it may be a way to kind of get a feel for the different odds of hitting different hands pre-flop based on your hole cards.

Hope this helps. (and is finally right) :)

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  • For your flush draw example there are three ways you can make your flush: hit the turn and river blank; turn blank and hit the river; or hit both the turn and the river. The other scenario is that you miss both. The odds of you hitting a non-flush making card are 38/47 on the turn and 37/46 on the river. The odds of you missing are then 38/47*37/46= 65%. The other 35% of the time you make a flush (16% you turn a flush, 16% you river a flush and 3% both turn and river). Same logic can be applied to straight draws. This doesn't answer the original q of how draws are hit on the flop itself
    – Chris
    Jul 1, 2015 at 13:19
  • You make many excellent point. I have edited my answer, can you weigh in on this? Am I doing that right? @Chris
    – Tech Savant
    Jul 1, 2015 at 13:37
  • It looks like you're trying to calculate the odds of flopping a flush, not a flush draw. In this case the odds of the events all happening is a multiplication of the odds of each individual event (11/50*10/49*9/48 so 0.84%) OP is asking the odds of flopping a flush draw, so 2 suited, one unsuited out of the three in all combinations. Have another go :)
    – Chris
    Jul 1, 2015 at 14:12
  • @Chris Thanks for your input on this, I have done another edit. Can you let me know what you think. :) Maybe I finally got it this time.
    – Tech Savant
    Jul 2, 2015 at 15:42
  • actually, I'm interested in a straight draw with at most one card the same suit as your suited connectors, that way you can add whatever that probability is to 10.94 (the probability of flopping a flush draw) for the probability of flopping either.
    – user3163829
    Jul 2, 2015 at 18:39
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The rule I always use and that is easy to understand is this :

Count your outs. To know your odds when you are on the flop and the turn is coming then multiply your outs by 2. If you have an open ended straight draw it's gonna be 8 outs * 2 = 16% to hit your out on the turn. If you want TURN+RIVER then it's times 4 (2 cards to come). so in this case if you both go all in on the flop and villain beats you then you have ~32% chance of hitting your outs.

Keep in mind that this is an approximation, however it's most of the time close from the real number. Here on the flop you have 34% real value(which is close from 32). On the turn with 1 card to come you have 19.89% (which is relatively close from 16%).

Using that you can calculate your odds on the fly. Also don't forget to withdraw possible outs if they are against you. ie: if you have 8d6h on 5s7sQh you have potentially 6 outs and not 8.

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I believe this is not a simple calculating task. with regards to flopping a flush or a flush draw - you need 2 out of 11 of your suit cards and 1 non-suit card, thus the number of such flops would be C(11,2)*C(39,1) whereas the total no of flops would be C(52,3).

This method avoids overcounting when you need to calculate the no of 3-suited flops (ready flush), n=C(11,3).

The next step would be ready straights and the open ended, double-belly and gut shot draws. After calculating all these, you need to subtract the flush draws and flushes to avoid overcounting and sum all the numbers and divide by C(50,3). That would be the percentage of favourable flops for a suited connector without taking into account what you opponent might hold and particular attention needs to be paid straight flopability of a connectorc This would be different from 98s to a 86s or 43s.

I will work on the subject too and will post my results, as I meant to do this for long time now.

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The probability of flopping either a straight draw or a flush draw is

1-(P(Not flopping either one))

1-(P(Not flopping either one)) = 1 - (1-0.109)*(1-0.104) = 1 - (0.891*0.896) = 1-0.798 = 0.202 = 20.2%.

So about 20%...

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