A box contains five black balls and one white ball. Alan and Bill take turns to draw a ball from the box, starting with Alan. The first boy to draw the white ball wins the game. Assuming that they do not replace the balls as they draw them out, find the probability that Bill wins the game. If the game is changed, so that, in the new game, they replace each ball after it has been drawn out, find the probabilities that:

(a) Alan wins at his first attempt;

(b) Alan wins at his second attempt;

(c) Alan wins at his third attempt. Show that these answers are terms in a Geometric Progression. Hence find the - probability that Alan wins the new game.