Your formula for the probability checks out.
This is what I am getting for the expectation
where the horizontal axis is the waiting time and the vertical is the expected time to wait. But it was done while I am half asleep and needs some checking.
The formula that produces this graph is the piecewise function:
Surprisingly this says that the expected waiting time is 10 after t >= 30.
If no one else can get it I will work on it tomorrow, I am exhausted from taking a long biking trip and having to return in the night through the forest. Much stress, much fear and much adrenalin seems to have drained the energy right out of me.
One thing still puzzles me, since I was interpreting the overlaps wrong I was getting the expectation wrong. How was I getting the probability right?
I think I see what your problem is... I do not know if your answer is wrong. You are misinterpreting what I was saying in that earlier post.
I was saying that
I never meant to imply that this formula is the solution to that problem. It may be or it may not. Having not worked on this problem at all I have no idea what the right answer is.