A sailing competition between two boats, A and B, consists of a series of independent races, the competition being won by the first boat to win three races. Every race is won by either A or B, and their respective probabilities of winning are influenced by the weather. In rough weather the probability that A will win is 0.9; in fine weather the probability that A will win is 0.4. For each race the weather is either rough or fine, the probability of rough weather being 0.2. Show that the probability that A will win the first race is 0.5. Given that the first race was won by A, determine the conditional probability that

(a) the weather for the first race was rough,

(b) A will win the competition. (C)