Question 45
Two boys $A$ and $B$ have a bag with $2009$ coloured balls: $2007$ balls are green and two are blue. They play a game with the following rules: When a boy gets the bag he draws two balls from it. If the two balls have the same colour he continues to draw one ball at a time until he draws a ball with the other colour than the first two drawn balls. At that time the bag is turned over to the other boy. If there is only one ball left in the bag, when it is turned over to a boy, he draws that ball. The game is over when the bag is empty. The winner is the boy with most balls when the bag is empty. What is the probability that $B$ wins if $A$ starts.