08-30-2023 , 12:25 PM
I was researching a specific Hold'em situation and came up with the following equation after some algebra.

Probability = (8*R^2 - 38*R+ 46.5)/1225

Can you guess what it represents (I'm assuming it is correct)?

Hint: R is hero's top rank in an unpaired hand

Spoiler:
It is the probability hero has the high rank unpaired hand against one opponent with a non-pair hand. A top rank tie is given 1/2 a "win".

It is useful for assessing the "goodness" of your hand. A Jack high is the approximate median goodness with a probability of 49%. It reduces to 23% against two opponents and to 11% against 3 opponents. In practice, adjustmemt should be made for playability, suitedness/conectedness, kicker rank, etc.
09-05-2023 , 03:22 PM
In case anyone is interested, here are the results for all Hero high ranks against 1, 2 and 3 opponents. Simulation was used for the 2 and 3 opponent cases.

09-06-2023 , 10:57 AM
Your numerator is slightly off but appears to be fixed by replacing 46.5 with 45. (Plus it wouldn't make sense for there to be half-combinations.) Also, the formula assumes zero information about Villain's hand, unlike what you said in the spoiler.

Two general formulas are:

C(4r-9, 2n) / C(50, 2n)

C(50-2n, 59-4r) / C(50, 59-4r)

where n=#villains and r is from 3 to 14 (since 2 is impossible with an unpaired hand).

With n=1:
Code:
``` 0.0024489795918367346
0.017142857142857144
0.044897959183673466
0.08571428571428572
0.13959183673469389
0.20653061224489797
0.28653061224489795
0.3795918367346939
0.4857142857142857
0.6048979591836735
0.7371428571428571
0.8824489795918368```
With n=2:
Code:
``` 0.0
0.00015197568389057752
0.0014329135909683021
0.005927051671732523
0.01683022145028224
0.038449848024316106
0.07620495006513243
0.13662613981762917
0.22735562310030394
0.357147199305254
0.5358662613981763
0.7744897959183673```
With n=3:
Code:
``` 0.0
4.405092286683406e-7
2.907360909211048e-5
0.00031496409849786355
0.001707413770318488
0.00635258358662614
0.01862787668258793
0.04633408219902207
0.10214527994361482
0.2053165058807982
0.3836491784502885
0.6757142857142857```
09-06-2023 , 02:22 PM
The relatively small differences may be due to different assumptions/definitions. It might be due to your reference to “half-combinations.”

I assumed a half win against one opponent occurs when both players have the same top rank and no pairs. Then the better hand as I defined it is the one with the higher kicker. Without any kicker data, I assumed hero has the better kicker half the time. Against 2 or 3 players, there may be more than one tie so I used hero’s win count to be 1/(1+qty. ties) which, admittingly, is not totally accurate but is of very small occurrence probability.

I agree that no assumption is made about hero's hand. The statement was not well written

Of course, a more accurate assessment would be to look at hero’s exact hand - high rank and kicker, much too complicated for me.
09-18-2023 , 05:58 PM
Oh ok, my formula is the probability that Hero has the highest rank, which doesn't include the chance of tying and out-kicking. It also doesn't exclude the chance of Villain having a lower pocket pair, which I'm guessing yours does (since those hands beat Hero's). But what I mean by half-combinations is that your denominator is C(50,2) and yet your numerator isn't an integer.

When heads-up, I believe the chance of having the highest rank OR out-kicking Villain is:

[C(4r-9, 2) + 6(r-3)] / C(50,2)

Code:
```0.0024489795918367346
0.022040816326530613
0.054693877551020405
0.10040816326530612
0.15918367346938775
0.2310204081632653
0.3159183673469388
0.4138775510204082
0.5248979591836734
0.6489795918367347
0.7861224489795918
0.9363265306122449```
If we exclude the chance of Villain having a lower pocket pair, it's actually the same as my original formula because we subtract 6(r-3) pocket pair combos, leaving us with a numerator of C(4r-9, 2).

More villains would make for some busywork and I don't doubt your sims.

m