Since this has come up a couple times on this board... Please correct any errors, I only have the equivalent of a minor in math. There are a lot of factors not included in this simplistic analysis such as straits,
quads, flushes, other
pair on board and bluffing.
Situation: You hold a small
pair (lets say 22) and another player comes out with a pre-
flop raise. You put him on a likely larger
pair.
Your plan:
Call his pre-
flop bet and look for a
set. Go nuclear if you get it, otherwise
fold.
How much money needs to be left in both of your stacks for this to be profitable, assuming you can hit him up for lots of additional chips if you hit your
set? Also assume you can't make the
lay down if he makes his
set too.
Odds of you not flopping a
set if your opponent doesn't have one of your cards (safe assumption if you have 88 or lower)
46/48 = 0.95833333333333333333333333333333
45/47 = 0.95744680851063829787234042553191
44/46 = 0.95652173913043478260869565217391
=
0.87765957446808510638297872340227
Inverse to get odds of flopping a
set
1 - 0.87765957446808510638297872340227 =
0.122340425531914893617021276598
or
1 in 8.1739130434782608695652173911537
Odds he doesn't get his
set if you do.
45/47 = 0.95744680851063829787234042553191
44/46 = 0.95652173913043478260869565217391
0.91581868640148011100832562442008
Flopping a
set and not losing to a better
set on the
turn +
river (doesn't factor in your
draw to
quads)
43/45 = 0.95555555555555555555555555555556
42/44 = 0.95454545454545454545454545454545
0.91212121212121212121212121212068
Algebra time.
Let:
X is pre-
flop betting
Y is
post-
flop betting
Z is the pot size = 2(X + Y)
we'll assume blinds and limpers
cover the
rake...
Cost for getting and playing that
set is:
8.17 X + Y
You'll pay X for the
flop 8.17 times to get your
set, then pay Y to get
showdown.
Profit on a win:
(0.916) * (0.912) * Z =
0.835 * Z
Odds of them not flopping a
set times odds of not getting beat on the
turn or
river.
Which yields this equation to determine our
break even point.
-8.17 X - Y + 0.835 Z = 0
8.17 X + Y = 0.835 Z
8.17 X + Y = 0.835 * 2(X + Y)
8.17 X + Y = 1.670(X + Y)
8.17 X + Y = 1.670 X + 1.670 Y
8.17 X = 1.670 X + 0.670 Y
6.5 X = 0.670 Y
6.5 X = 0.670 Y
9.7 X = Y
Adding in some room for profit and fuge factor, you need at least
10x the pre-
flop raise worth of chips in BOTH stacks that you think you can get in the pot if you hit your
set to
call that bet. You probably can subtract a little for each additional caller, but not quite 1x because of the additional chance of a better 3 of a kind,
flush or
straight.