3-card deck cont.

S = the size of the pot.
b = the size of the bet.
p = probability Bill's hand holds up.
(1-p) = prob Adam is dealt a higher card.
Now the 2 X 2 matrix looks like this.

______\_______BILL
ADAM	Call	Fold
B/B	p(S+b)-(1-p)b	S
B/CH	p(S+b)-0	pS

Reset the payoffs to Bill's point of view.
The 'B/B X Fold' cell is zero.
Bill is guaranteed zero by always folding.

The 'B/B X Call' cell is now
(1-p)(S+b)-pb
Bill wins the pot plus a bet(S+b) whenever Adam
is bluffing. Adam is bluffing (1-p) part
of the time. Bill loses one bet p part of
the times.

Bill's wins must be greater than his losses else
it's always right to fold.
1a. (1-p)(S+b)>pb
_____S+b-pS-pb>pb
_____S+b>pS+2pb
_____S+b>p(S+2b)
lb. p < (S+b)/(S+2b)
p must be smaller than the fraction for Bill
to ever correctly call. p can be larger by there
being too many outs. But usually p can only
be larger by Adam bluffing too infrequently.

If the pot size is large relative to the bet
size as in fixed limit this expression(1b) will
approach 1. If the bet size is significantly
large relative to the pot size the fraction
approaches 1/2.

Adam is a low frequency bluffer. Bill should
never call. Whenever Bill never calls, Adam has
some free bluffs. As long as p remains above
the fraction (S+b)/(S+2b) Bill should not call.
Adam should control himself. Never show his bluffs.
This will allow Adam many more free bluffs.