Ok, after seeing your math now I get the 60% number. For some reason when I read your initial posts I thought, "Shouldn't that be 67%?" but didn't really know what you were looking at.
The break even point is exactly 2/3 or 67%.
Eq 1: 2*z - 3*(1-z)
Eq 2: z - (1-z)
Set eq 1 = eq 2 and solve for z.
Now, if you want to investigate the effects of the small percentage of the time that we get 3-bet on the river by a hand we beat then the equations will be:
X = percentage we are ahead but he won’t 3-bet our check raise
Y = percentage we are ahead and he 3-bet’s our check raise
Eq 3: 2x + 3y – 3*(1-x-y)
Eq 4: x+y – (1-x-y)
Set 3 = 4 gives us:
3x + 4y = 2
or:
x = 2/3 – 4/3y
y = ˝ - 3/4x
we would need to assume percentages for either or x or y and solve for the other quantity.
Example:
So if we are ahead 60% (x) of the time he needs to 3-bet us with a worse hand 5% (y) of the time for the c/r to be equal to c/c.
Here is a chart of how this breaks down:
<font class="small">Code:</font><hr /><pre>
x y x+y
0.5 0.125 0.625
0.51 0.1175 0.6275
0.52 0.11 0.63
0.53 0.1025 0.6325
0.54 0.095 0.635
0.55 0.0875 0.6375
0.56 0.08 0.64
0.57 0.0725 0.6425
0.58 0.065 0.645
0.59 0.0575 0.6475
0.6 0.05 0.65
0.61 0.0425 0.6525
0.62 0.035 0.655
0.63 0.0275 0.6575
0.64 0.02 0.66
0.65 0.0125 0.6625
0.66 0.005 0.665
</pre><hr />
edit:
this really makes me want to c/c instead of c/r (I was never in favor of c/r'ing).
It is a little harder to add an analysis of betting out and calling a raise compared to simply checking.