Using pot equity to attempt to derive a formula

[JMBills]

When it comes to Poker Theory and probabilities I am a neophyte but I want to become better at thoroughly understanding the mathematics involved. Here's one particular situation I am trying to derive a formula from.

Say you're at a no-limit hold'em game with an opponent who will all-in you with ANY pocket pair, preflop, and you act after him/her. There aren't any other opponents who will call him when he all-ins you. Your hole cards are AJs.

I used Pokerstove to find out that your average equity with AJs against any pocket pair is 43.579%. I'm not entirely sure if I ran the simulation correctly but for now let's assume that you will win 43.579% of the time against his holdings.

I'm thinking that in order for your call to be profitable:

CallAmount <= .43579(Bigblind + small blind + opponents bet + your call)

In a game where the small blind is .5x the big blind,we can think of it as:

CallAmount <= .43579(1.5Bigblind + 2callamount)
or
CallAmount <= .653685 blinds + .87158callamount in order to be profitable.

Here's where I'm stumped... somehow this algebraic equation must be balanced and we can find the ratio of the blind to your call amount in order for your call to be profitable. Can anybody speculate or guide me to other resources to find an answer?

[garion888]

1. Subtract .87158 from both sides...
2. Use the distributive property.
[ QUOTE ]
Callamount - .87158*Callamount = Callamount*(1 - .87158)

[/ QUOTE ]
3. Divide both sides by (1 - .87158) or .12842
4. You now have your answer. It should be
5.09021*Big Blind

[jdl22]

Firstly you are also assuming here that your opponent will not put you all in with any other hands. I'm assuming below that you're in neither blind.

Let x = stack size in number of big blinds (it's actually the min of your stack and your opponent's stack)

Pot odds are 1.5+x:x. This is (1.5+x)/x:1. The odds against you winning are (1-.43579)/.43579:1 or about 1.3:1. To make you indifferent set x so that (1.5+x)/x=1.3 as it turns out x=5.

The other way to do it is to use probabilities. If you win you win x+1.5. If you lose then you lose x. Hence you need x so that .43579(x+1.5)-.56421x = 0. This also gives you x=5 approximately (rounding to the nearest tenth).