A childhood belief about weeing and pooing

I thought girls peed out of a little hairy stick that popped out of their bottom when they needed to go.
This was because I had been taken into the ladies when Iwas about 4 and had stood beside the toilet while my mum had crouched over the pan and had a pee.
What I had seen was her peeing and her pubic hair caught up in the flow and when she stopped it disappeared back up into her bum.
I thought that this stick popped out at the moment you started to pee and then disappeared again when you finished but how did it know when to pop out?
I thought this was the way girls peed until I was about 9 and saw a girl take her pants down in front of me and pee.