A train and a bus randomly arrive at a mutual station between 9:00 A.M. and 10:00 A.M. The train stays at the station for 10 minutes upon its arrival, and the bus stays at the station for "x" minutes upon its arrival. How long does the bus have to stay at the station for the probability of the train and bus meeting each other to equal exactly 50%?

Good luck. This gets a much higher difficulty level than that dumb Car and Goat one I previously posted. If any of you get it, I'll be very impressed. And guesses will not be accepted, you must show your work, or explain it at least.

Hint: The answer is actually a pretty in-depth one. The use of formulas and the knowledge of probability and critical thinking is next to required.