Does it makes sense to say that in HU if hero holds Q,Q then there's a 4.24% chance villain is ahead after a J,7,2 flop excluding draws?

Villain pockets that put him ahead:
JJ, 77, 22, J7, J2, 72, AA, KK

Probs using a 47-card (post-flop) universe are:
JJ, 77, 22 each are .002775
J7, J2, 72 each are .008326
AA, KK each are .004525

Sum = .042352 or 4.24%

The short answer is "no, it doesn't make sense".

The thing you should think about is what equity your hand (or your range) has against his range.

I agree.

First figure out which hands "should" be in your range depending on the individual equities. Then figure out how you should play your range depending on the range vs range equities.

The 4.24% number (assuming you've done the maths correctly) would only make sense if villain sees the flop with 100% of hands. In the real world (especially in a multiplayer game, not HU) villain is unlikely to be seeing the flop with 72 or J2 or even J7.
Build a real range for villain and then use Flopzilla or Equilab or Combonator to find out how often he can beat top pair on J72r. It's not very often, especially if his pre-flop action was "call" (meaning he can't have QQ+, unless he's a passive whale).

EDIT: As a quick example, if you opened QQ UTG and villain was in the BB and called pre with a nitty range like JJ-22, AQs-A2s, KTs+, QTs+, JTs, T9s, 98s, 87s, 76s, 65s, AQo-ATo, KJo+, QJo (he 3-bets QQ+/AK), then on J72r, he has 9 combos of sets representing 5.3% of his range. He never has two pairs or an overpair, so you'd be ahead about 95% of the time.

You are answering the wrong question. the question you are answering is:

"What is the likelihood that I am ahead if I hold XX, Villain is playing any two cards preflop, and action stops on the flop?"

Your math is correct in analyzing this question, but this question is completely meaningless from a poker standpoint.

I guess we need to back up a abit and ask, what problem or situation are you trying to analyze?

Is the next step when calculating equity to determine probability of winning? I.e. if villain is ahead calculate probability hero hits his outs, or if hero is ahead calculate 1 minus probability villain hits his outs?

For simplicity let's put villain on a specific hand: 55

Therefore villain has 2 outs: fives

To determine prob hero wins, would we calculate total probability of following scenarios then subtract from 1?
1. Turn is a 5, river no help to hero
2. Turn is a 5, river is a 5
3. Turn is no help to hero, river is a five

On the turn, there are 43 out of 45 cards that do not help villain. On the River, there are 42 cards out of 44 cards that do not help villain.

Likelihood turn does not help villain, 43/45. If turn does not help villain, likelihood that river does not help villain is 43/44. To figure your total likelihood of winning, multiply the probability of both of those events together.

Continuing with a simplistic example, let's put villain on following range: 66,55,44.

Is prob hero wins (39/45) x (39/44) = .768 ?

If so, what about the fact it's really only 2 outs that are applicable to each hand in the range. Villain has either 66 OR 55 OR 44. So not ALL six outs will actually help. Do we apply any discounting or adjustments to calculation to account for this?

No matter which of the possible hands he has, he has only two outs to win. There's no difference between him having back 4s or red 6s.

So you calculate his probability to hit his outs on the flop or/and on the river.

Your equation isn't quite right and it's a little more complicated.

Hero wins if villain completely misses or villain can hit one of his 2 outs on one street but hero hits a queen on the other.

P(H wins) = P(V doesn’t hit his outs on turn and river) + P(V hits an out on one street and H hits a Q on the other)

= (43/45)*(42/44) + 2*(2/45)*(2/44) = 0.916

This checks with what Equilab calculates.

Understood, so since each scenario has 2 outs we use (43/45 * 43/44).

What if villain's range contains hands with differing numbers of outs?

For example, if we add KJ to villain's range then we have:

66 --> 2 outs
55 --> 2 outs
44 --> 2 outs
KJ --> 5 outs (3K, 2J)

Would we use 5 outs in prob calculation, i.e. (40/45)*(40/44), or apply some kind of weighting to each of the hands in his range to determine the number of outs we use in calculation.

You are on the right track. When you calculate your likelihood of winning against a range, you do a weighted average. In your example above, if you think his range as 66, 55, and 44 (and only those hands), you multply the likelihood of winning against each hand (91%) times the likelihood of having that hand (33.3%), and then add them up.

Let's take a much more meaningful example, a common preflop situation. You have raised with QQ., and a tight player has three bet all in. You believe that he would only do that with AK, AA, KK. You want to figure your likelhood of winning.

First,you look at your percentages against each hand
AK - 57%
AA - 18%
KK - 18%

Then you need to weight them by their frequency. We know there are 16 combos to make AK, 6 combos to make AA, and 6 combos to make KK

So, you will end up with the hand the following pecentage of time
AK -57%
AA - 21.5%
KK - 21.5%

to figure your percentage of win against the range, you now take a weighted average of the win percentage of each hand in the range

probability against the range = (57%)*(57%) + (18%)*(21.5%)+(18%)*(21.5%)= 40.2%

Your equity for QQ versus a range of AK, AA, KK is around 40%

You would have 2 different equations, one with 2 outs and one with 5 outs for a villain win. Yes, then you would weight the results by the occurrence frequencies.

For the pairs, there are 3 *6 = 18 combos and for KJ, there are 4*3=12 combos.

The KJ equation is P(H wins) = (40/45)*(39/44) = 0.788

Overall, H wins with probability (18*0.916 +12*0.788 )/30 = 0.864

Since the KJ combo, if suited, can give villain a win with a flush, this result differs in the third place from Equilab depending on what suit interactions you allow. You can actually model that, but we should not worry about such a small difference.

So, essentially, the steps for determining prob of winning are as follows?

1) Establish villain’s range
2) Group different hole-card combinations by number of outs and tally count in each group
3) Multiple each count by applicable weight, where weight = 1 - prob(villain does not win)
4) Sum products.

Is that correct?


And to clarify,

For the pairs: 3 x 6 = 18 is derived from (4 cards choose 2) = 6 then 6 x 3 pairs = 18, yes?

And for KJ: 4 x 3 = 12 is derived from (4 kings choose 1) = 4 and (3 jacks remaining choose 1) = 3 then 4 x 3 = 12, yes?

This is correct. To be honest, very rarely are you going to be calculating this during play. Work with an equity calculator to memorize the equity in common scenerios. You are rarely going to need to be exact, so being able to estimate your equity against villain's probable range is good enough.

Here is another scenario where I’ve tried to work out the probability of winning to help crystallize my understanding. My probability math skills are not exactly top notch which is why I’m working on them.

Despite there being apps out there to calculate these probabilities, I’m attempting to understand the actual core calculations.

Kindly have a look.

Hero: 99
Board after flop: 234
Villain Range TT+, ATs+, AJo+, KJs+

Probabilities for hero winning are,

Subrange 1: AA/KK/QQ/JJ/TT
Hero lands 9, no villain set = (2/45)*(42/44) + (43/45)*(2/44) = .0424 + .0434 = .086

Subrange 2: AKo/AQo/AJo/AKd/AQd/AJd/ATd/KJd/KQd
No villain pair = (39/45)*(38/44)=.748

Subrange 3: AKc/AKs/AKh/AQc/AQs/AQh/AQh/AJc/AJs/AJh/ATc/ATs/ATh/ KQc/KQs/KQh/KJc/KJs/KJh
No villain pair, no villain flush draw = (30/45)*(29/44) = .439
No villain pair, busted villain flush = (9/45)*(30/44) + (30/45)*(9/44) = .136 + .136 = .272
Total = .439 + .272 = .711

Combos:
Subrange 1: (4c2)*5 = 30
Subrange 2: (4c1)*(4c1)*3 + 6 = 54
Subrange 3: (3c1)(3c1)*6 = 54
Total combos = 138

Prob(Hero wins) = (30/138 * .086) + (54/138 * .748) + (54/138 * .711) = .019 + .293 + .278 = .59

59% hero wins? Feels high, no?

Equilab shows 45.4%, so you have some mistakes.

I did a quick look and note that your combo counts are wrong for non-pairs. Off suit is 12 combos each and you have 16. Suited is 4 combos each and you have 9. Also you didn’t consider the possibility of villain hitting a wheel straight. I didn’t look further.