I had my argument all lined up and was just sitting down to write as a proof, but then I thought, hey I'll just do a test case just to prove it. I computed all the different possible cases, calculated the number of stops for each case, and calculated the expected value. Lo and behold your formula nailed it.
Not that you probably care, but where I think I goofed up was thinking of this more like a random process. Instead of thinking that each passenger randomly picks a stop at the beginning, I was thinking of each passenger going to a stop and deciding whether to get off or not, with probability . But that is not a good model of what is happening. For one thing, to work it this way, the probabilities of getting off would have to increase to 1, and not stay constant at ("last stop, everybody off). I interpreted
as the probability that of k people on the bus that they don't get off at that stop. And my thought went that there were only k people on the bus near the beginning and certainly not that many toward the end. So it was this formula that I was questioning.
I humbly submit that you did this problem correctly.
But darn it, you should see the formula I had come up with! It is correct, but it is ugly.
And no, it's not hard to swallow:
a) I am quite happy with . In fact it is very useful as this case points out. I never questioned it.
b) I absolutely don't mind being wrong, I just didn't believe I was wrong!