A chess player playing with white pieces will win any game with probability 0.4, draw with probability 0.4 and lose with probability 0.2. The same probabilities when she plays with black pieces are 0.2, 0.5 and 0.3. Before each game a fair coin is tossed to decide whether she will play with the white pieces or the black pieces. Hence the probability she plays with either colour pieces is 0.5.

i. Given that she won her last game, what is the probability she was playing with the white pieces? (4 marks)

ii. Given that she did not lose either of her last two games, what is the probability she was playing with the same colour of pieces in both games? (6 marks)

iii. Let N denote the number of wins she will score in the next 10 games. Calculate E(N) and Var(N). (4 marks)