Odds of Floping a Straight Draw

[farf]

Hi all. Somehow I found myself in a bit of an argument with a friend over the odds. We are trying to figure out what the chances of floping a straight draw with connected cards, and what the mathematics are behind it. I have an answer but I cannot seem to find anywhere to substansiate it. I was hoping someone here could help me.

More specifically, if you have a 9-T, what is the chance you will flop a straight draw.

The answer I came up with is 10.51972789%

Looking forward to hearing your answers.

Farf

[Phil153]

You need to flop one of five scenarios:

78x, where x is not a J or a 6;
8Jx, where is is not a Q or 7;
QJx, where x is not an K or 8
68Q, (a double gutshot)
7JK, (a double gutshot)

These are mutually exclusive, and the first three have the same probability. For the first one, the probability is given by:

P(887) + P(778) + P(78x) = [C(4,2)*C(4,1) + C(4,2)*C(4,1) + 4*4*34]/C(50,3)
= [24 + 24 + 544]/C(50,3) = 592/19600 = 3.02%

Multiplying by 3 gives 9.06% chance of flopping just an OESD. For the double gutshots, we have:

4*4*4 = 64 ways each, times 2 gives 128 combinations. 128/19600 = 0.65%

So I get a total 9.71% chance of flopping a straight draw. After calculating I just looked through BruceZ's old posts and he claims 9.6% for this figure. So I don't know where the discrepancy is.

[BruceZ]

[ QUOTE ]
You need to flop one of five scenarios:

78x, where x is not a J or a 6;
8Jx, where is is not a Q or 7;
QJx, where x is not an K or 8
68Q, (a double gutshot)
7JK, (a double gutshot)

These are mutually exclusive, and the first three have the same probability. For the first one, the probability is given by:

P(887) + P(778) + P(78x) = [C(4,2)*C(4,1) + C(4,2)*C(4,1) + 4*4*34]/C(50,3)
= [24 + 24 + 544]/C(50,3) = 592/19600 = 3.02%

Multiplying by 3 gives 9.06% chance of flopping just an OESD. For the double gutshots, we have:

4*4*4 = 64 ways each, times 2 gives 128 combinations. 128/19600 = 0.65%

So I get a total 9.71% chance of flopping a straight draw. After calculating I just looked through BruceZ's old posts and he claims 9.6% for this figure. So I don't know where the discrepancy is.

[/ QUOTE ]

The difference is that I am assuming a suited connector and excluding made flushes. Otherwise our calculations are exactly the same. Here are the details of my calculation. I also got 9.71% for the offsuit case here, just as you did.

The OPs number is very close to the value one gets by not carefully separating out the paired flops, hence double counting them, which is a very a common error.