I thought of a problem while waiting at a railroad crossing. Very simply, it goes like this:
When you approach a railroad crossing (with a train crossing it), how many cars do you want to see waiting there?
Assuming, of course, that you want to spend as little time waiting at the intersection as possible, there are arguments to be made for fewer and for more cars. If you see fewer cars, it’s likelier that the total time the train will have spent at the crossing will be small. If there are more cars, it’s likelier that the train has already spent most of its total time at the intersection. So it’s a bit of a paradox, right?
For the sake of this problem, assume that you’re not able to measure how fast or how long the given train is. However, you have two values to work with: R and f(t). R is the number of cars per unit time which approach the crossing, which is assumed to be constant. f(t) is the continuous probability distribution that a train will spend a total time t at the intersection, given that you encountered it at the intersection. (Without that last given condition, the problem would look different, and probably harder.)
I haven’t spent too much time on this problem, but I think it might be a little tricky. If anyone has any thoughts on it, I’d love to hear them!
(Also, let’s say that you consider only the cars on your side of the tracks.)