Being mildly sad - prayer can work instantly.
Feeling a bit bleugh - often provides short term relief.
Colds, indigestion, blisters - often works after a few days.
Bit of a bad back - can feel a bit better.
Legs of different lengths - instant, but often pointless. Often reverts (or even reverses) after walking around a bit.
Impotence - try to avoid laying-on of hands. Or, on second thoughts... no, let's leave it.
Piles - definitely avoid laying-on of hands.
Depression - may not work as well as you think.
Lost limbs - works very infrequently.
Based on this article. Basically, at the Beaker Folk we believe prayer can work miracles. And medicine can do everyday, scientifically-predicable healing which can be pretty special. Why not try both? (But be warned - God is incredibly reliable at working through physical laws, but a bit random on the "apparently impossible" stuff).