Timed Extractor Fan Circuit - how does this work?

[Bluffer]

Let's say I connect the positive probe to pin 4 on the LM324 (the IC +ve supply) and the negative probe to pin 11 on the LM324 (the IC gnd pin).

It looks to me like the positive probe has a constantly positive dc voltage around 7V relative to earth and the negative probe will see a 240V AC voltage relative to earth.

 

[betwixt]

You are measuring it correctly. It's only the voltage across the IC that matters.

Be very careful not to confuse "gnd" or "ground" which is local to the circuit and is the negative side of the supply with "Earth" which is an assumed zero voltage benath your feet.
True "Earth" whether from a mains supply Earth wire or from a rod in the ground plays no part in this circuits operation whatsoever and it should not be conected to any part at all. The circuit 'floats' across the neutral and live wires and has no outside reference to anything else. In fact, as it operates on AC, you could swap the live and neutral wires and it would perform exactly the same.

Brian.