I am not here with purpose to produce long useless posts, contrary to what some superficial posters think i do it to offer better insight into the math of the situation almost always. So do not take my response as nitty, only as an insight into the conclusion that 25% cannot ever be her range unless we are some supernit idiot that doesnt understand basic things or unless its the bubble and we need the money near a major life changing pay jump etc.
So yes OP's example was from CO with 3 left to act but his question was from sb so it makes a difference. And the opponent we want folding is a different more respectable super deep player that risk from our 10.5bb means nothing for her 90bb. The proper range from CO in his first example with 8bb is more like;
39.4%, 22+ Ax+ K2s+ K7o+ Q6s+ Q9o+ J7s+ JTo T7s+ T9o 97s+ 86s+ 76s 65s
the bb calling 37.0%, 22+ Ax+ K2s+ K6o+ Q6s+ Q9o+ J8s+ JTo T8s+
These numbers do not need a big adjustment like our case due to removal effects of 7 having folded.
In any case a player like our BB here knows from this that A7o is already in our range at CO. This indicates at the very least
28.2%, 22+ A2s+ A7o+ A5o K6s+ KTo+ Q8s+ QTo+ J8s+ JTo T8s+ 97s+ 87s 76s
from that position if we werent at 39% anyway which cant be excluded.
That right there is important to her and a lesson not to show our hands when with small stack at such spots unless the image created helps us later (eg show an AJs or 88 here why not).
It hints to her that if our range was 28% at CO then at sb its easily higher. Not only i did show that even 100% gives us on avg a small profit regardless how she reacts but she knows we tried 28% before and look like easily 35-40% here if we remain consistent even at the tightest bias.
Since she only needs however 41.3% equity vs our range as i showed in #4 she will easily have that vs a tight perceived 40% range with at least 33% of hands. So the smallest she can have is 33% and even that is too small because to derive it she has to assume that our A7o in the previous example is the smallest equity AX we were pushing something she cant know. Not only this is a statistical extreme assumption but even the 29% to 40% jump from CO to SB is conservative when it ought to be trivially obvious to anyone that 50% for pushing range is already safe regardless how the opponent reacts.
Why? Well for one we already saw above that even 32o while negative EV push is not by a lot. So hands significantly better than 32o but still lower than 50% will do even better and pass the push test. That clearly forces the pushing range over 50% and below 100% hence a rough avg near 70% is ok even with simple calcs . But sure live people dont have time to think or cant calculate such things ( thats why my posts are long to do it here instead and know then or extrapolate by having some insight from some basic stacks like 5-10-15-20).
Additional reasons now why his range should be at the very least 50% with still easy calculations. Well because if she reacts over 50% he gains because he has better range and dead chips from antes. If she reacts less than 50% he gains because even say a 30% vs 50% range leaves him 45% avg equity enough to cover the gains from her folding too much. Basically there is no hand in say the bottom of 50% that combined with a fold equity of some 50% from her isnt leading to overall gain as push. Examples k4o,J7o,J4s. These hands as pushes with 50% fold equity yield 42.5,38,39% respectively of 21.765 bb when called and 12.265 when she folds or (j4s say) 0.5*12.265+0.5*0.39*21.765=10.38>10.5-0.5=10bb=folding.
Doing this instantly shows why the bottom of 50% range hands pass the test easily.
Basically its elementary for any solid veteran player LAG or TAG and even some tight newbie ones (we leave out the loose gamblers and bullies and bad LAGs because these land on...correct by accident here lol) that it has to be at least 50% and the refinement is really between 50 and 70%. And without being a nit any choice between 50 and 85% is doing a reasonable job at being practically unexploitable.
In any case she will need to assume we have at east 50% and then the above calculation that yielded 33% for her rises easily to include all the hands that pass the 41% req. vs a 50% range at least or ; 45% 22+,A2s+,K2s+,Q2s+,J7s+,T8s+,98s,A2o+,K2o+,Q8o+,J9 o+,T9o.
This is why she cant have less than 45% here. Our range is harder to guess but her range is definitely over 45% simply because ours while remaining harder to refine is definitely over 50% and 45% of hands have at least 41% vs a 50% range. Obviously vs wider range more hands will qualify hence the trend toward 50% for her too.
Noway in hell a solid player with very deep stack without a specific read or a bubble pay jump type scared money for us hint is calling us tighter than 50% here.
But try a tight player with less experience or a situation that before this hand we had a stream of exposed very top hands or hints of scared money nitty behavior and yes assuming they call us tighter than 50% is a possibility. Not with the offered data in this table though. As always poker advice must fit the situation not what generically holds true elsewhere. I am sure elsewhere 25% calling frequency fits tight inexperienced players or scared stacks near pay jumps.
So 50-70% vs good opponents and even wider vs others unless they are loose gamblers that obliviously call us just for the hell of it.