Technically, you haven't said ... so we have to guess ... whether your bulbs
are arranged in a series circuit or a parallel circuit.
I've read your question, then read it again, then read it between the lines,
then read it between the words, then read it between the letters. I THINK
that when you say they're connected "one after the other", you're describing
all three bulbs connected in one line. There's no place in the circuit where
the path splits, no 'fork in the road', and no place where the current has to
decide which turn to take.
-- That's called a "series circuit".
-- Since there's only one path for current to take, all the way around the
circuit from one battery terminal to the other one, every electron has to
flow through every bulb.
-- If a gap opens up anywhere in the circuit, then there's no way for current
to make the trip, and everything stops.
-- If one bulb lights up, then all the bulbs light up.
-- And if one bulb burns out or gets stolen, then all the bulbs go out.
