Calling river bet, 2 possible villain hands

[Jouster777]

The following was triggered by a specific example in one of the NL Hold EM boards. I attempt to provide the general solution below.

If the range of villain’s hands is limited to 2 possibilities, you have a sense of the relative likelihood of each of these possibilities, and you are faced with a raise on the river then the following formula will determine when to, at minimum, call.

BASIC START:
EV = P(win)*(amount in pot with raise) – P(lose)*(amount to call)
If we set EV to 0 then…
P(win)*(amount in pot with raise) = P(lose)*(amount to call)
P(win)/P(lose) = (amount to call)/(amount in pot with raise)

r = (amount to call)/(amount in pot with raise) => pot odds
r is assumed always <1, i.e. something in pot before villain’s bet

P(win)/P(lose) = r (intuitively obvious)


ACTUAL SOLUTION:
x = P(hand 1)
1-x = P(hand2)
y = P(hero wins when villain has hand 1)
z = P(hero wins when villain has hand 2) (y≠z)


P(win) = xy+(1-x)z
= xy + z - xz
P(lose) = x(1-y)+(1-x)(1-z)
= x – xy + 1 – z – x +xz
= 1- xy – z + xz
= 1-P(win)

Placing this into P(win)/P(lose) = r yields
(xy + z – xz)/(1- xy – z + xz) = r
this simplifies to:

x = [r/(1+r) – z]/(y-z)

If the lower win probability hand is always hand #1 then:
1. Fold if the likelihood that villain has hand #1 more than the derived percent from this equation
2. Call if likelihood that villain has hand #1 less often than the derived percent from this equation
3. Fold to a negative number, call to a number >1

Probably not real convenient to remember but easily put in a spreadsheet etc. for quick calculations

I am a new poster. Is this type of thing useful for this forum? Too simple? Any criticisms welcome.