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Calling with a Small Pair

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…

Click here to continue reading the full thread, Calling a Pre-Flop Raise with the Smaller Pair.

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