This is a weird thought experiment that came into my head today. I apologize for it being very "informal."
Say I am driving down a road that is lit on both sides by streetlamps.
As I'm driving, I notice one of the streetlamps go out as I drive past it. It is only in my field of view for a short time, so I don't see what happens after that.
Now, the question: is it more likely that the streetlamp I saw turns off and on repeatedly at some interval (say, due to a faulty bulb), and I happened to see just one of those events?
Or is it more likely that the bulb stays on all the time, and I just happened to see the moment it burnt out?
In my head, I visualize the problem as such:
Here's what the streetlamp's status would look like if it is blinking on and off (O is on and X is off)
[... O O O X X O O O X X O O O X X O O O ...]
I observe one of these intervals at random, and see it go from O to X. Since this series of on/off is infinitely long, I could appear at any moment and would be likely to see some switch.
If it ISN'T a faulty bulb, and I happen to see it burn out permanently for the first time, the interval would look like this:
[... O O O O O O O O O O O O X X X X ...]
It is much less likely that I see the on/off switch, since that only happens at one point in the infinitely long timeline.
So it should be more likely that the bulb is blinking on and off.
Is that true? Am I wrong? Is this "streetlamp paradox" more complicated than how I visualized it in my head?
I apologize if this is more abstract than what this stackexchange normally sees -- it was the only stackexchange to come up when I searched "statistics."
EDIT: This problem seems to change if when I observed the streetlamp as I drove past, it was off the whole time, instead of switching from on to off. In this case, I would assume it is burnt out completely, since even though the "on to off" switch only happens one time in the timeline of the burnt-out bulb, the "off" segment happens for infinity.