At dusk you want to turn on your headlights when half of the oncoming drivers have theirs on, judging only...

Problem

At dusk you want to turn on your headlights when half of the oncoming
drivers have theirs on, judging only by runs of oncoming cars with
lights on; how long a run should be your trigger?

[ Editor: the title sentence above is a summary; the original posting
follows. ]

We're driving along a highway.  Lots of traffic coming the other
way, volume roughly constant.  It's getting toward dusk.  Pretty
soon, more and more of the oncoming cars have their lights on.
We want to turn *our* lights on just at the point when half of the
drivers have put their lights on.  To be difficult, we want to
guess when this point is by counting strings of oncoming cars
with their lights on, and switching ours on when we get
to N.  An oncoming car with lights off resets the count to zero.
What's the choice of N which gives the best chance of achieving
the 50% criteria?  Does N depend on the volume of oncoming
traffic?  (Surely yes.)

This is a fairly vague wording of the problem, I admit.  It's
possible to pin down what's meant by the 50% point a little better:
Start counting all cars (C), and cars with headlights on (H),
well before dusk, at time t=0.  At some time when it's getting
pretty dark, one will find a time t=T when H = C/2.  The 50% time,
in retrospect, was t=T/2.  We assume that the volume of oncoming
traffic was roughly constant throughout the interval.

The goal, then, is to define how many oncoming cars in a row, all
with headlights on, should be the trigger for me to turn on my
lights at a time which most closely estimates T/2.

-- D. F. Munro <{big machine}!aluxp!aluxa!dfm>