Straight or Flush from Pre-flop to river

[magates]

First, please forgive the rather crude and probably incorrect math involved. My education stopped at college algebra and even that was 10 years ago. I've tried to hack an answer to my question based on other posts in this forum, but I'm far from certain that I did it correctly. Any help would be appreciated.

What I'm trying to figure out is: what percentage of the time will you complete a flush, a straight, or a straight flush by the river when you start with 76s? I found the basic question asked by searching, but I only found the chance of flopping a straight or a flush draw, not the chance of completing one by the river.

With some searching, I've found the following:

Your chance of flopping a clean 8-out straight draw is 7.21%
Your chance of flopping a 4-flush is 10.94%
Your chance of flopping a straight (w/o a 3-flush board) is 1.22%
Your chance of flopping a flush is 0.84%

If your chance of flopping a flush draw is 10.94%, and your chance of completing a flush draw from the flop to the river is 35%, then your chance of both flopping a flush draw and completing a flush is 11%*35%=3.9%. Is that right? Assuming it is, I'll continue . . . .

Your chance of completing a flush draw by the river, or flopping a flush is 3.9%+0.84%=4.74%?

Your chance of flopping an 8-out straight draw and completing by the river is 7.21%*32%=2.3%

Your chance of completing an 8-out straight draw or flopping a clean straight (w/o a 3-flush board) is 2.3%+1.22%=3.52%

Thus your chance of either flopping a straight or flush, or completing one or the other by the river is 3.52%+4.74%=8.26%

Is that approximately correct?

What is your chance of flopping both a straight and a flush draw, e.g. 2h 5c 8h? What is your chance of flopping a straight-flush or a straight-flush draw? The two combined probably don't add much to your pre-flop equity, but I'm curious.

[BruceZ]

[ QUOTE ]
First, please forgive the rather crude and probably incorrect math involved. My education stopped at college algebra and even that was 10 years ago. I've tried to hack an answer to my question based on other posts in this forum, but I'm far from certain that I did it correctly. Any help would be appreciated.

What I'm trying to figure out is: what percentage of the time will you complete a flush, a straight, or a straight flush by the river when you start with 76s? I found the basic question asked by searching, but I only found the chance of flopping a straight or a flush draw, not the chance of completing one by the river.

With some searching, I've found the following:

Your chance of flopping a clean 8-out straight draw is 7.21%
Your chance of flopping a 4-flush is 10.94%
Your chance of flopping a straight (w/o a 3-flush board) is 1.22%
Your chance of flopping a flush is 0.84%

If your chance of flopping a flush draw is 10.94%, and your chance of completing a flush draw from the flop to the river is 35%, then your chance of both flopping a flush draw and completing a flush is 11%*35%=3.9%. Is that right? Assuming it is, I'll continue . . . .

Your chance of completing a flush draw by the river, or flopping a flush is 3.9%+0.84%=4.74%?

Your chance of flopping an 8-out straight draw and completing by the river is 7.21%*32%=2.3%

Your chance of completing an 8-out straight draw or flopping a clean straight (w/o a 3-flush board) is 2.3%+1.22%=3.52%

Thus your chance of either flopping a straight or flush, or completing one or the other by the river is 3.52%+4.74%=8.26%

Is that approximately correct?

[/ QUOTE ]

Good job using search and trying to figure it out for yourself. Here is a useful link that has other things you may be interested in. Follow the links. Here are my comments:

1. 7.21% does not count flops with a straight draw and a paired board, or with 3 of the same suit. Perhaps you wanted to ignore these flops since they are dangerous, but you didn't ignore paired boards when you did the flush draws. You do want to exclude flush draws here since they are already being counted by the 10.94%. So if you want the probability of a straight draw with no flush draws, but including paired boards and 3 of the same suit, that number is 8.36%.

2. 1.22% also excludes cases where the 3 board cards are the same suit. If you wanted to include these, that number is 1.3%.

3. There are other ways to make a flush or straight by the river that you are not counting, such as flopping and completing a gut shot, back door flush, or 3 to a straight.

4. Your calculations look fine.


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What is your chance of flopping both a straight and a flush draw, e.g. 2h 5c 8h?

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Including the straight flush draw, 1.2%.


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What is your chance of flopping a straight-flush or a straight-flush draw?

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straight flush: 4/19600 = 0.02%.

straight flush draw: 3*33/19600 =~ 0.51%.
(3 for 54h, 85h, 98h, times 48-15=33 other cards that don't complete flush or straight).