Two archers A and B shoot alternately at at target until one of them hits the centre of the target and is declared the winner. Independently, A and B have probabilities of and, respectively, of hitting the centre of the target on each occasion they shoot.

(a) Given that A shoots first, find (i) the probability that A wins on his second shot, (ii) the probability that A wins on his third shot, (iii) the probability that A wins.

(b) Given that the archers toss a fair coin to determine who shoots first, find the probability that A wins. (NEAB)