Katie is one of ten members of a social club. Each week one member is selected at random to win a prize.
i. What is the probability that in the first 7 weeks Katie will win at least 1 prize?
ii. Show that in the first 20 weeks Katie has a greater chance of winning exactly 2 prizes than of winning exactly 1 prize.
iii. For how many weeks must Katie participate in the prize drawing so that she has a greater chance of winning exactly 3 prizes than of winning exactly 2 prizes?