Math!

[Pokey]

[ QUOTE ]
i usually and regrettably ask Pokey

[/ QUOTE ]

Speak the devil's name and he appears.

The easiest way to do this is to plug all the information into PokerStove and see what it spits out:
<font class="small">Code:</font><hr /><pre>
Text results appended to pokerstove.txt

207,900 games 0.063 secs 3,300,000 games/sec

Board: Tc 9d 2h
Dead:

equity (%) win (%) tie (%)
Hand 1: 55.1804 % 53.75% 01.43% { QQ }
Hand 2: 44.8196 % 43.39% 01.43% { JJ+, AKs, AKo }
</pre><hr />
See where it says that QQ has 55.18% equity against villain's range? That means that for every dollar that enters the pot, hero will, on average, collect 55.2 cents. Since any called wager in a heads-up pot involves putting in 50 cents to build the pot $1, this means hero increases his profits by pushing this flop and getting called.

Specifically, after pushing the pot will have $49 + $89 + $89 = $227, of which hero will win (on average) $125.26. Since hero must put $89 into the pot to build this, hero's pot equity after the flop push is $125.26 - $89 = $36.26 more than his flop investment.

If hero does NOT push, the pot remains at $49, and hero wins 55.18% of that money, meaning hero's pot equity after checking the flop is $27.04. In other words, pushing gains hero an extra $9.22. (Note that this is 5.18% of the $178 that goes into the pot on the flop bet -- hero's profit margin on the heads-up bet is 5.18%, since hero wins the pot 55.18% of the time, and 50% would be a break-even situation.)

Things get more complicated when you include the possibility that villain folds to a push, when you include turn and river betting rounds, etc., but for the basic problem presented, this works nicely.

In general, if you have more than 50% pot equity and zero folding equity, betting is +EV heads-up. If you have more than 33.3% pot equity and zero folding equity, betting is +EV in a two-way pot. Etc.

[Dennisa]

Does JJ and AK really call 100% of the time a push from you? Also can you really rule out tt and 99 in your assumption?

Text results appended to pokerstove.txt

243,540 games 0.094 secs 2,590,851 games/sec

Board: Tc 9d 2h
Dead:

equity (%) win (%) tie (%)
Hand 1: 51.1431 % 49.92% 01.22% { 99+, AKs, AKo }
Hand 2: 48.8569 % 47.64% 01.22% { QQ }

[Saborion]

[ QUOTE ]
Does JJ and AK really call 100% of the time a push from you? Also can you really rule out tt and 99 in your assumption?

[/ QUOTE ]
That's irrelevant. I wasn't interested in whether they always call or not, I was interested in how to do EV calculations. The answer to Question 3 , where he folds JJ a certain percent of the time, shows me how to do those kind of calculations as well, which was all I needed for now.

[Saborion]

Pokey, I know I'm +EV if he always call with AK and JJ, but given that he rarely will call with AK, and that he sometimes, although rarely, will fold JJ, I can't use Stove. That's the reason for my intial post; I need to learn how to adjust for those percentages where he folds. Using Stove to find out what kind of equity I have vs a certain hand in his range if very nice, but I do need to do some manual calculations after that.