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Let's say I open from the cut off for 3.5x in any 6max game under 50NL. I probably raise about 27% of my hands from the cut off which looks something like:
22+,A2s+,K9s+,Q9s+,J9s+,T8s+,96s+,86s+,75s+,65s,A7 o+,K9o+,QTo+,J9o+
Let's assume the BB is your average 45-55/5/1.0 fish, who loves to defend his big blind, we'll say38%:
AcAd,KcKh,QcQs,QcQh,JJ-22,AcKc,AdKd,AcQc,AdQd,AJs-A2s,K3s+,Q8s+,J8s+,T8s+,97s+,86s+,76s,65s,54s, half the AKo combos,AQo-A2o,K8o+,Q9o+,J9o+,T9o
If the board is:
Ad 6s 9h
If they check, their range changes because there are certain hands in that range this player wouldn't check. Or wouldn't check often enough for it to be practical to consider.
If they bet their range changes, a large number of these hands will not bet the flop if they miss OOP. If they bet it narrows their range, because they won't donk 100% of their range when defending on this board (although whether or not they should is debatable).
I guess it comes down to the whole "it depends", and poker is very situational. But I'd like to believe (and at some point I will run simulations to back this up) that there are more transition states in hold 'em that result in ranges being altered than result in them staying the same (excluding transitions which result in the completion of the hand).
I guess the general rule is that the aggressiveness or passivity of the opposing player will dictate whether his range is more greatly affected by betting, checking, calling, or check calling.
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