Think of it this way. The elevator could be on
floor 1 moving up
floor 2 moving up
floor 2 moving down
floor 3 moving up
floor 3 moving down
floor 4 moving up
floor 4 moving down
5 up
5 down
6 up
6 down
7 up
7 down
8 up
8 down
9 up
9 down
10 up
10 down
11 up
11 down
12 up
12 down
All of these would mean that the guy on the 13th floor would get on an elevator that was moving up.
The rest
13 moving up
13 moving down
14 moving up
14 moving down
15moving down
13 up is still moving up but the other 4 possibilities is the only way for the elevator to be coming down when he gets on.
So there are 28 possibilities and only 4 where the guy is lucky. So the probability that he gets an elevator coming down is 4/28 or 1/7....leaving 6/7 that the elevator is moving up.
As for the second part of your question, Since the elevators are independent and you are asking only about one of the elevators, each one of those elevators moves up 6/7 of the time, regardless of how many elevators.
Now if the question said that you had 7 elevators stop on the floor all at the same time, the probability would say that 1 of the seven would be going down.
Hope this helped.
