[Buzz]

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The degeneracy is different

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Sam - I don't know what that statement means. I don't know what "degeneracy" used in that context means.

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6[(19/48)*(9/47)*(20/46)]=0.1977

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That's correct, and concise, but not exactly how I probably would have thought of it. (But I'm not suggesting that my path of reasoning is better than yours. Indeed, yours looks more direct).

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OK, so the conclusion so far is this: AQ42 flops an uncounterfeited nut low draw 17.6% of the time, and a once-couterfeited low draw 19.8% of the time. In that 19.8% of the time in which the draw is once-counterfeited, 6.6% of the time it will be the 4, and the draw will still be to the nut low. The other 13.2% of the time it will pair the A or 2, making the draw to the second-best low. Given all of that, AQ42 flops a nut low draw 24.2% of the time and a second-nut low draw 13.2% of the time. Is this all correct?

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Somehow I have the feeling that you know more about this stuff than I do and that I'm like a country bumpkin struggling along with something he's pretty unsophisticated about, but O.K., I'll bite.

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AQ42 flops an uncounterfeited nut low draw 17.6% of the time

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10*(2*4*6+4*4*19) = 3520

or 20*16*19/2 + 5*6*16 = 3520

3520 is the number of three-card combinations (not permutations) that would result in a flop where you had an uncounterfeited low draw, if holding A24Q. All of those are nut low draws.

The number of possible flops is C(48,3), or 17296.

3520/17296 = 0.2035. That's the probability of flopping an uncounterfeited low draw. Expressed as a percentage that would be 20.35%.

I had previously written:
(19*20*16/2)/(48*47*46/6) =
3040/17296 = 0.1758

As I look at it now, it seems I left paired flops out of that computation.

My error. Sorry.

It should have read:
(19*20*16/2)+(5*6*16)/(48*47*46/6) =
(3040+480)/17296 =
3520/17296 = 0.2035

Therefore AQ42 flops an uncounterfeited nut low draw 20.35% of the time. I think I've got it right this time.

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and a once-couterfeited low draw 19.8% of the time.

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•One flop card would be an ace, deuce of four (any of nine cards from your perspective, holding A24Q),
•a second flop card would be a trey, five, six, seven or eight (any of 20 cards from your perspective, holding A24Q),
•and the third flop card would be a high card (any of 19 cards from your perspective, holding A24Q).

O.K., here goes:

(9*20*19)/17296 = 3420/17296 = 0.1977

So, yeah that one looks correct.

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In that 19.8% of the time in which the draw is once-counterfeited, 6.6% of the time it will be the 4, and the draw will still be to the nut low. The other 13.2% of the time it will pair the A or 2, making the draw to the second-best low.

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Yes. Looks like 6.6 and 13.2 is the correct partitioning of 19.8 to get a 3 to 6 ratio (or a 1 to 2 ratio).

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Given all of that, AQ42 flops a nut low draw 24.2% of the time and a second-nut low draw 13.2% of the time. Is this all correct?

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No. If the 0.1758 was correct, it would be. (17.58+6.6 = 24.2)

However, the 0.1758 is incorrect. (Sorry, my error).

Instead, 20.35+19.77/3 = 26.94

A24Q flops a nut low draw 26.94% of the time and a second-nut low draw 13.18% of the time.

It also directly flops (no draw) the nut low 6.48% and the second nut low 5.55% of the time.

It also flops various other favorable hands and draws. (I'm not sure where you're going with this).

I usually check my numbers backwards an forwards, as many ways as I can check them. I didn't this time because I got the same result as Mike Cappelletti on my first try. Also, my numnbers added up to 100%. Rats! What a can of worms! There must be something else wrong with that post too, since the numbers added up to 100% using the (erroneous) 17.58%. I'll go back and look at that post when I get back from the game tonight. I'll probably post a correction.

I guess Mike and I both omitted flops such as 3-3-5 from our tabulations. Mike's a nice guy and tries to be helpful in his writings. I find his stuff very readable and interesting. Don't fault him if there's a small glitch here or there. It's easy to go wrong in making these tedious tabulations. Lots of places to make errors.

Just my opinion.

Buzz

P.S. Below is the scratch math for the 6.48% and 5.55%.
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10*4*4*4 = 640
Another way to think of it is:
20*16*12/6 = 640 (checks)

10*3*4*4 = 480
Another way to think of it is:
3*20*16/2 = 480 (checks)

(640+480)/17296 = 1120/17296 = 0.0648
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6*10*4*4 = 960
Another way to think of it is:
6*20*16/2 = 960 (checks)
then 960/17296 = 0.555






Whew. How did I get myself into this? Gets tedious after a while. Easy to make a mistake by leaving something out.