Hi guys, first time poster, please be easy on me!
I did a bit of theory today since i seemed to be running into this situation a bit more often than I expected. If anyone has some input I would gladly appreciate it!

Why are you posting a picture?

Your section "based on equity" is wrong, because you can not compare equity to a one-street decision. Your equity is your chance of winning by the river, but you're talking about making a single-street call with 3 streets of betting left. If you want to use equity to make a decision, you need to use "effective" pot odds, which is your estimated *total* investment vs your estimated total pot to win.

I did not look at your calculations closely for the 2nd part - but you'll find that they've been done on this forum and others many times already.

Okay, I think I understand what you mean in terms of the equity. It did seem a bit strange to me, but I think you clarified it.

In regards to finding the calculations I did do a good number of searches to try to find some information, but none of the articles I found had very good evidence and mostly just "yes you should do that" or "no thats not what I do". Sorry if this is indeed a double post

The reason I posted a picture is because I did it all in word beforehand and the formatting didn't paste correctly onto the forum

I like to use the old 5% of effective stacks rule of thumb for set mining with some mods.
So, generally, I want to take a set mine only if the cost to call (an open or a 3 bet, doesn't matter much) is no more than 1/20th the money I could win. That is money in pot + effective stacks.

That would be a pure set mine (small pair with little chance of winning the pot unimproved at show down).

But I would be willing to take less implied odds based on how favorable the following factors are;
Position
Villain's post flop tendencies (will he get sticky with an over pair or TPTK? Will he get aggro and bluff off a bunch of chips when you flop your set? I want to set mine vs players who make a lot of calling mistakes post flop or players who make a lot of aggro mistakes post flop. The middle is not that good for us and TAGs, in particular, are tough to set mine against.
Strength of villains range (I think it's best to flat pocket pairs against very weak and very strong ranges but generally the middle strength ranges are not good to set mine against. Vs strong range we have better implied odds, vs weak range we will sometimes get to show down and win unimproved and we can find some bluffs vs weaker ranges. Also, strong ranges tend to make big but not huge hands like TPTK and over pairs but middle strength ranges can have a lot of suited connectors and suited broadways, maybe Axs that can put us on the business end of a cooler when a lot of money goes in post flop or at least we can say middle strength ranges will tend to be drawing live against us post flop when stacks get in relative to ranges like AQ+,JJ+ that will often have 2 outs to beat us when we flop a set.
Size of pocket pair. The ability to over set a player once in a great while makes some difference and the built in show down value of 88 makes it better than a pure set mine with 22 because we can sometimes peel a flop bet and bink our 2 outer and sometimes we can peel the c-bet and check down and win.
Other players in the pot. If another player comes in between us and the opener then there are more sources of implied odds and better express odds and a set will beat two opponents almost as often as one opponent so it can sweeten the deal a bit, be careful about counting bluff oppertunities and show down value of a medium pocket pair and then ALSO counting the extra implied odd potential of a thru caller because our "66" is less likely to win at show down against two players unimproved and it's not usually going to be great to turn a pp into a 2 out bluff against multiple opponents but the presence of another player in the pot is usually going to be a good thing for you set mines on balance.
Capping the action. If we are considering a set mine out of the BB we know we won't be squeezed out after VPIP'ing.
Position. It's much easier for villains to pot control vs your flopped sets when we are OOP and so being in position helps us squeeze max value out and it can let us get take a free one off or get all the way to show down some of the time.

I'm sure I'm leaving somethings out but, basically, if I am;
OOP vs a decent TAG who has opened from CO (has a lot of stuff like A5s, QJs, JTs, 99, in his range compared to relatively few JJ+,AQ+), and I have a player yet to act (maybe I'm SB) who could squeeze me out. And if I have a pair of 3's, for instance, I'm not going to be real happy to set mine unless I am getting pretty much 20 to 1 potential implied odds on my call.

But,
If I am in position with 77 vs a player with a tight range pre-flop who gets pretty sticky and pretty aggro post flop and if I know he is likely to c-bet his entire range on most flops and then shut down unless he has at least a good TP but bet bet bet and maybe bet bet bet call with his TPTK+ then I might just go ahead and flick in the call if it is nearing 10% effective stacks.

Kinda like an updated version of the rule of 5 and 10.
Just remember that the 5 or 10 is % of EFFECTIVE stacks, remember to include money already in the pot not just stacks, and adjust based on factors that actually make sense instead of "I was in that 5 to 10 ballpark and I hadn't played a hand in a while so.." or "It was only 7% of my stack so I could afford to gamble"


Otherwise the math and the profitability or value of the situation doesn't change much with a 3 bet compared to an open raise. It's still an implied odds estimation with other considerations dragged in to evaluate the overall value of the spot.

There's not really any precise math you can do, because each pair has a different value (lower pairs get oversetted or counterfeited more often, for example) and being IP or OOP can have a major effect on your ability to get paid in full when you bink.

A general rule of thumb I used to use was based on implied odds. To whit: Calculate what the stacks behind will be if you call the 3-bet, and compare it to the price of calling. If you're IP, odds of 10 to 1 should be good. If you're OOP, you'd be better off sticking with 15 to 1.

e.g. Assume 100bb deep.
You open for 3bb with 77 in CO. BB 3-bets to 10bb. If you call the additional 7bb, the effective stack will be 90bb. 10:1 implied odds IP only needs 10 * 7bb = 70bb, so this should be a profitable set-mine.

If, you're OOP (CO vs a 3-bet by BTN), then ideally you want 15:1 odds. 15 * 7bb = 105bb, so this won't be profitable. (You'll only have 90bb for post-flop if you call the 3-bet).

It's a bit rough and ready, but it gets you in roughly the right area. If you find yourself in some spots that look particularly close, there's no shame in folding the smallest pairs (55-22). These tend to do fairly badly in the long run, as they play so horribly when they miss the flop.

The calculation you made there was only for value. For value, I think somewhere around the 15 x the size of the call as potential winnings is good to start with... Also, you have presumed that we always lose value when we miss, but sometimes the villain won't bet etc etc. Sometimes we can miss and reach the river with the better hand... And you have presumed that we always win or lose stacks when we hit, but that isn't the case either.

You haven't mentioned making the villain fold at all... If the range you've given him is correct, we can probably get all kinds of folds. This is player dependant, but, this guy is only really aiming for high boards and he doesn't have many Ax in his range. This means that we could take down many of the Axx boards and many of the low-med boards as standard, but we do need to have some guess as to how he will react to our aggression before we can make that assumption... He has basically left our own boards for us. What I mean by this, is that if the flop comes 567 it will usually smash our perceived range and he will have almost no hands in the race. Against a half decent player, you could get loads folds on those boards. This might well mean that we don't need so many of those low pairs in our actual range. If we expect to take the low boards down without reaching showdown those low hands lose favour. And so we might be better off replacing some of them in our actual range with hands like KQ, TJ, as these could destroy the villains range for value on a nice board... Or, he could be the type of guy who won't even fold his missed AK at a later date. In this case, we would be encouraged to play lower cards for value.

Yeah, even using this 15x standard is still just a rudimentary technique. We need to look to our perceived range and the villains range and consider the future cards and see how much our actual hand will make off the villains range for value and how often we can bluff him, before we're able to see if this will be a profitable play.

Couple of good posts before mine! Don't worry about rusty. She hit the menopause recently n has been grumpy ever since.