Saving Electricity

Let's say I'm living in a house, and in that house is a bathroom. Nothing amazing about this, but I hope you have an ounce or two of imagination to handle this. Now in this bathroom is a light bulb, which is connected to the grid of the house. Like most light bulbs, you can turn it on when you flip the switch assigned to it. To get a bit more technical, you actually allow electricity to flow through the little spirally things (that sounds really technical) and they glow to make light. I'm not a very advanced electrician, but just correct me if I'm wrong and if you don't know better, just assume I'm right.

Now what if that light bulb busts? Just to clarify: the glass is still intact, but it just doesn't work anymore. You turn on the switch, and when you walk into the bathroom, it's like "what? I can't see anything!"

The big question comes now. I'm hoping some smart people read this, so that they can answer it in the comments below. If the light bulb is busted, but the switch is still on, does it use up electricity?

It appears not to, because it's still dark in said bathroom, but I've heard people complaining about it, and it doesn't make sense. In my opinion (not factually based) the light bulb should still be glowing (lighting up the bathroom) if it was actually using up electricity. The second school of thought goes like this: the light bulb is broken, but the path for electricity is still open, so it doesn't matter if you're using it productively, you're still paying for it.

I would really like some insight on this situation. I really need to take a crap, and I'm not sure whether to leave the switch on or not.