Odds to flop a draw?

[Net Warrior]

Can someone help me figure out the chances of flopping a draw, either a 4-str8 or a 4-flush, when you have a meduim suited connected hands like 87s? Thanks in advance.

[BruceZ]

We had a thread about this recently. Here is my solution with a few typos corrected. The answer is 19.3%. I am very confident in this answer because the same answer was obtained by a completely different and much more complicated method. The following method produces the answer with a minimal number of steps. I am assuming we have JTs instead of 87s, and this is equivalent.

First find the probability of flopping exactly a 4-flush draw without the cards for a straight draw. We can always flop a 4-flush draw without the straight draw cards if we flop a 4-flush draw with no more than 1 of the 4 denominations 8,9,Q, or K (more than one of the same denomination is OK). We can also always do it when we flop only the denominations 8,K. We can also always do it if we flop only 8,Q or 9,K so long as we don’t flop exactly 8,Q,A or 7,9,K, since these are the only double belly buster draws. These are all the cases, since the other combinations of two cards 8,9; 9,Q; and QK are straight draws. So now we count the cases.

No 8,9,Q,K: C(7,2)*27 = 567
Exactly 1 denomination of 8,9,Q,K: 4*7*30 + C(7,2)*12 = 1092
8,K: 2*3*7 + 1*33 = 75
8,Q but not 8,Q,A: 2*3*6 + 1*30 = 66
9,K but not 7,9,K: same as above = 66

Where I have added two terms, the first is for exactly 1 of the listed cards being a flush card. The second term is for either exactly 0 or 2 of the listed cards being flush cards, as appropriate to the case. Exactly 1 denominaton means multiple of same denomination still OK.

(567 + 1092 + 75 + 66 + 66)/C(50,3) = 9.5%. This is the probability of a clean flush draw.

Now the probability of a flush draw including the times we also make a straight draw or straight is C(11,2)*39/C(50,3) = 10.9%. So the probability of making a flush draw while at the same time also making either a straight draw or straight is 10.9% - 9.5% = 1.4%. The probability of making a straight draw or straight including the times we also make a flush draw is [ 3(12*42 + 4*33 + 6*4*2) + 4*4*4*2-2 ]/C(50,3) = 11.1%. Therefore, the probability of a straight draw or a straight is 11.1% - 1.4% = 9.7%. The probability of a straight including straight-flush is 4*4*4*4/C(50,3) = 1.3%, so the probability of a clean straight draw is 9.7% - 1.3% = 8.4%. The probability of a straight draw or a flush draw or both is 9.5% + 11.1% - 1.3% = <font color="red">19.3%</font color> which is what you wanted to know.

So in summary:

P(clean flush draw) = 9.5%
P(clean straight draw) = 8.4%
P(clean straight draw OR clean flush draw not both) = 17.9%
<font color="red">P(clean straight draw OR clean flush draw OR both) = 19.3%</font color>
P(both clean straight AND clean flush draw) = 1.4%

[Net Warrior]

Bruce,
I've been reading through some of your other posts recently and was hoping you'd respond to this one. Now I see that this problem was more complicated then I realized. I've reread your answer several times but need to go over it some more. So, bottem line, it's about 4.2 to 1 to flop an 8 outer or better(I think). Thanks again.

[BruceZ]

Right, 4.2-1. It is complicated mainly because the exact answer requires that we compute the probability of flopping just a flush draw by itself without a straight draw, so that we can add that to the probability of flopping a straight draw which may include a flush draw, 9.5% + (11.1%-1.3%) = 19.3%. It is relatively easy to compute the probabilities of a straight draw which may include a flush draw, or a flush draw which may include a straight draw similar to what I did in the 2nd to the last paragraph. We can't just add these two together or we would be double counting the times we flop both draws. If we did add these numbers our answer would be off by 1.4%, so we would get 3.8-1 instead of 4.2-1. That's still a reasonable approximation, and a whole lot easier to compute than the exact answer as is often the case. For the exact answer, we had to compute the probability of making at least one of these draws without making the other draw, and then we can get the other one by subtracting as I did. I chose to compute the probability of a flush draw by itself rather than a straight draw by itself as this looked easier. I did a little more arithmetic at the end than was necessary to just answer your question because I was also interested in getting the probability of flopping just a straight draw by itself without the flush draw, since we get this almost for free once we've come this far.

[Copernicus]

H'eA gets 19.8% for 87s, close enough not to figure out why there is a difference. You may also want to add 2.23% for completed straight or better hands flopped.

[BruceZ]

WHO gets 19.8%? If this is from a book, please give the page number. I have never seen this exact stat quoted anywhere.

[Copernicus]

H'eA = Hold'em Analyzer (software)

[ccwhoelse?]

middle size connectors would give you a straight draw 8.4% of the time but what about higher and lower connectors that can make less then four possible straights?