I am having a very hard time understanding when exactly a transistor turns on. For example in the following circuit from Everycircuit:
Why does the transistor turn on when the push button is off, but not when it's on? The only way it seems to make sense to me is that the voltage relative to the base is "more positive" when the resistance to from the base to the positive terminal is lower than the resistance from the base to the negative terminal. For example, if I change the resistances:
From this, how would current even have anything to do with 'turning on' the transistor at all? It seems like it's only related to voltage from the below. Could someone please describe why exactly the transistor is being turned on in these conditions? Not just conceptually but with equations as well, if possible to show the current/voltage going in and such.