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Actually, the probability is 100% that the elevator will be going in the
right direction. Proof by induction:
N=1. Trivially true, since both you and the elevator only have one
floor to go to.
Assume true for N, prove for N+1:
If you are on any of the first N floors, then it is true by the
induction hypothesis. If you are on the N+1st floor, then both you
and the elevator have only one choice, namely down. Therefore,
it is true for all N+1 floors.
QED.
right direction. Proof by induction:
N=1. Trivially true, since both you and the elevator only have one
floor to go to.
Assume true for N, prove for N+1:
If you are on any of the first N floors, then it is true by the
induction hypothesis. If you are on the N+1st floor, then both you
and the elevator have only one choice, namely down. Therefore,
it is true for all N+1 floors.
QED.
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