I was reading Advanced PLO by Hwang and this line made me want to know how often we do flop a 13 out straight draw when holding JT98 (suits don't matter)


How can I calculate the percentage of flops those four different ways represent when compared to the total number of possible flops?


I can't math very well but I'm tweaking so I felt like I had to give it a shot, freestyling math for science, so I did this (possible) atrocity on notepad just now; It feels wrong though.

is that the right way?

Just for clarity, are you looking for the joint probability of flopping a nut straight draw?

That is, a draw and not a nut straight?

And, the joint prob, not the four individual probs (that would then need to be disentangled)?

Finally, do you want to exclude a monotone flop where you know that you would not be drawing to the nuts (flush>straight)?

That's correct. The joint probability of flopping a 13 out nut straight draw with a hand like JT98, in the four Flop examples from Hwang; not a straight.

Good point, I need to exclude monotone flops. I'm not sure how.

Well, using that logic you'd want to exclude paired flops too since, of course, you'd not be drawing to the nuts in that case as well.

There is no "trick" to doing this. You just have to write down all the possible flops which would give you a nut straight draw making sure not to count the same flop more than once. For example, 87x and 76x could "overlap" if the first X is a 6 and the second X is an 8.

In this problem it seems that you should be able to get out a piece of paper and simply write down all the possibilities.

Your approach in the original post seemed generally correct. But make sure to think carefully about what "X" can be. And take into account suits to avoid monotone flops.

For the first type of flop 87x, it might help if you split it into two cases: (i) 8 and 7 are same suit; (ii) 8 and 7 are different suits. Then in each case tally what the third card could be.

Do same for the other flop types. Make sure to not write down the same flop twice.

There are "easier" ways to do this but I think it is worthwhile to do it the manual, brute-force way first. Then you will be able to see any short-cuts you could apply.

Feel free to keep us posted with questions/comments/status/etc.