The Unmarked Bus Stop
This summer I lived in New York City for the first time since 2000. I've lived in New Jersey and New Hampshire since then. Both the City and I have changed enough for me not to remember my way around very well. That got me into trouble a few times -- I would think I knew where I was going, until I was sure I didn't. Once, after getting lost, I found myself waiting at an unfamiliar bus stop hoping to cut back across the city to where I was supposed to meet someone for dinner.
The N.Y.C. bus stop signs -- see an example above -- don't list the bus schedule. This is frustrating. Unless you're a local, you really have no sense of how long you will have to wait. What if that bus line isn't even running today? Yet, after you've been waiting some amount of time, the fact that you've waited has some value as information. You should conclude that, in general, buses are less frequent than you believed when you showed up. But you might also conclude that the coming bus will arrive sooner now than when you had arrived, given that you've been waiting some time without a bus.
Yes, all of this was running through my head as I waited at the bus stop. Let's call it the "unmarked bus stop problem": How long should you expect to wait after you've been waiting at a bus stop? The problem is a basic application of probability theory, but it's Christmastime and the solution is easy but practical.
The actual statement of the problem: Assume that buses show up at your stop according to a Poisson process with rate λ, which is itself unknown. Assume that, initially, all rates are equally likely so that you have a flat prior. As a function of the time you have already waited t, what is the expectation of the amount of time left before the arrival of your bus?
You can make this problem a bit more challenging, too. Almost always, when you get to a bus stop, you're not the only one waiting. What do you do with the information that others are waiting? Is it a sign that buses come infrequently? Or the bus is about to arrive?
Second problem: Assume that bus riders show up at the stop according to a Poisson process with rate μ, itself unknown with a flat prior. Assume that the arrival processes are independent. Assume that, when a bus arrives, all those waiting can always board. How long should you expect to wait given that there are n other people waiting there when you arrive and N waiting after wait-time t? Note: I haven't actually written this one out.
I will post the answer in the comments in a few days.