Can someone please calculate the probability of flopping an OESD with the following hands:

23 through QK
24 through KJ
25 through KT

I will tell you how to do it.

First, count the number of two-card combinations that form an OESD. With 23, for example, it has to be 45. But with 67, it could be 45, 58 or 89. With KT the only two-card combination is QJ, but you could also get three cards, say, 789. I assume you don't count that.

A specific two-card combination will appear in the flop 9 times out of 245, or 3.67%. It's approximately correct to multiply the number of combinations by 3.67%. It's not exact, because you double count time times when you flop a straight.

The 3.7% rule seems helpful -- how have you confirmed this?

The problem is that if we multiply this by 3 for the medium connector case (like JT) we get about 11.1%, whereas the correct probability of flopping an OESD is about 9.1% assuming that we want to exclude all made straights. This is because the probability of flopping a made straight with a medium connector is about 1.3%, and multiplying 3.7% by 3 not only counts these straights but double counts half of these straights, so that we are off by 1.5 times 1.3% or about 2%.

If you want a fraction that you can use this way, I suggest instead of 3.7% = 9/245 that you use 37/1225 =~ 3.02%. This does not count the made straights at all, so you can multiply it by the number of 2-card combinations without error. It does not exclude made flushes, but that only adds about 0.1% when the hole cards are suited.

Note that this does not include double gutshots, which are also 8-outs. It is not possible to use a simple fraction to include these since the number of them does not simply scale with the number of 2-card combinations. For the probability of flopping an OESD or a double gutshot, see the calculation explained in this thread for connectors JT-54.

The probability changes for QJs/43s, and for KQs/32s, and you can find the calculation of these probabilities in this post.

For 1-gappers QTs-53s, see the paragraph in blue in this post.

thanks - very helpful