Equations to Estimate Showdown Equity
Using multiple regression, I developed equations to estimate your showdown equity against various villain types for bets and calls. There are two independent variables: Hero Hand (X1) and Villain Playing Characteristic (X2) defined as follows:
Equation For Estimating Showdown Equity, Villain Bets (R=0.964)
Equity = 0.622 - 0.048 * X1 + 0.158 * X2
To possibly use within rules while playing
Equity = 0.6 - 0.05 * X1 + 0.2 * X2
Example: Hero has AJ against a moderate villain who bets.
X1 = 2. X2 = 25% Equity = 0.622 -0.048*2 + 1.58*0.25 =0.57
Equilab value for same inputs= 56%. Here, an excellent fit.
Approximation: Equity = 0.6 – 0.05*2+0.2*0.25 = 0.55
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Equation For Estimating Showdown Equity, Villain Calls (R=0.948)
Equity = 0.574 – 0.046 * X1 + 0275 * X2
To possibly use within rules while playing
Equity = 0.58 - 0.05 * X1 + 0.3 * X2
Example: You hold a type 3 hand (e.g., 66) and make a standard bet. A loose villain calls.
X1 = 3. X2 = 30%. Approx. Equity = 0.58 – 0.05 * 3 + 0.3 * 0.30 = 0.52
The exact value is 0.48
In most cases, the deviation from exact is less than or = 0.04. No account was made for suitedness except for types 7 and 8, so add 1% equity for suited and subtract 1% for offsuit. You can use fractional values for X1 and X2 to better represent the boundary cases.
Use at your discretion.
3 Replies
With the terrific response I got to my initial posting, here we go again:
Doing some research on pair probabilities, I came up with a simple linear equation to predict the equity of pair rank R against a random hand (R^2 = 99.64%).
Reducing it to 3 decimal places, it follows:
Equity = 0.465 + 0.027 * R
Example: For a pair of Jacks, R=11, Eq = 0.465 + 0.027*11 = 76.2%
Equiilab shows the equity to be 77.5%
For a pair of 8’s, predict vs actual is 68.1% vs 69.2%
For a pair of fours, 57.3% vs 57.0%
HIGH CARD ADVANTAGE. I know you've been waiting for this, so here it is:
If hero hand is a non-pair with high card of rank R > 3, the probability one opponent doesn't have a pair or a higher ranked card follows: (high card ties = 1/2 “win”)
Pr. = [C(x, 2)*16 + 18x + 4.5]/1225
where x = R-3
Example: Hero has 9y, y<9. x= 9 - 3= 6
Pr. = [C(6,2)*16+18*6+4.5)]/1225=28.8%
So what?
Well, the result says there is nearly a 30% chance you have a better hand than your opponent, ignoring suitedness and connectedness. If you held queen high, the probability is about 60%
Useful? You decide.
Ax Equations
Being dealt Ax is good, especially if kicker x is not too low, say > 8. Below are equations for estimating the probability none of N opponents have an ace with equal or better kicker than you hold. Ties are counted as half a “win”. Simplified regression equations were derived from simulations.
PREDICTION EQUATIONS x = Kicker, 2 to 13
Opps = 1: Pr= 0.87 + 0 .01x
Opps = 2: Pr= 0.74 + 0.02x
Opps = 3: Pr= 0.63 + 0.03x
Opps = 4: Pr= 0.52 + .034x
Example: Pr(neither of 2 villains has an Ace with kicker equal to or greater than 8 is
0.74 + 0.02*8 =90%.
Simulation showed 89.2%.
