The exercise is as follows:
Starting at 5:00 A.M., every half hour there is a flight from San Francisco airport to Los Angeles International airport. Suppose that none of these planes is completely sold out and that they always have room for passengers. A person who wants to fly to L.A. arrives at the airport at a random time between 8:45 A.M. and 9:45 A.M. Find the probability that she waits (a) at most 10 minutes; (b) at least 15 minutes.
The first sentence of the solution says:
Now the passenger waits at most 10 minutes if she arrives between 8:50 and 9:00 or 9:20 and 9:30; that is,
if $5 < X < 15$ or $35 < X < 45$
I agree with this procedure IF we only consider the cases where no matter what, the passenger won't wait more than 10 minutes. But the passenger could very well not wait for longer than 10 minutes if he arrives at ANY time, so I feel like we are excluding a lot of cases here. Where am I going wrong? Or, much more unlikely, where is the author going wrong?
Any help would be greatly appreciated because probability has been, is, and will forever be a nemesis of mine!
EDIT
It seems as though I understood the first sentence as saying that every $30$ minutes starting at 5:00 A.M., there is exactly one flight that takes off WITHIN the $30$ minutes time interval. It seems I am supposed to understand it as flights take off at exactly the beginning of the hour and exactly in middle of the hour. I don't think either interpretation is incorrect given the language.