How does a transistor behave when the gate is disconnected?
But what about when the gate is disconnected?


The disconneced gate pin acts as an antenna, and will pick up some electromagnetic noise from the environment - likely the 50 or 60 Hz from the nearest wall power lines. 

The end result is largely random, and there are other effects like leakage currents to account for. 

Thats why you want a pullup or pulldown resistor in cases where the gate of a MOSFet could otherwise be "open" - so your circuit is kept in a known state.