Two duelists, A and B , take alternate shots at each other, and the duel is over when a shot (fatal or otherwise!) hits its target. Each shot fired by A has a probability α of hitting B , and each shot fired by B has a probability β of hitting
A . Calculate the probabilities P 1 and P 2 , defined as follows, that A will win such a duel: P 1 , A fires the first shot; P 2 , B fires the first shot.
If they agree to fire simultaneously, rather than alternately, what is the probability P 3 that A will win, i.e. hit B without being hit himself?