The World Series is a postseason play-offseries between the two respective champions of the two major professional baseball leagues in North America: the American League (AL) and the National League (NL), which together constitute Major League Baseball. The World Series is a best-of-seven playoff; i.e., it consists of at most 7 games and the first team to win 4 games wins the championship. It is not necessary for the games to be won consecutively and draws are not permitted. Once a winner is decided, the Series ends.

a) Suppose the two teams in the World Series have equal probability of winning each of the 7 games and that the games are independent.

i. Calculate the probability that the World Series ends in 4 games.

ii. Calculate the probability that the World Series ends in 5 games.

iii. Calculate the probability that the World Series ends in 6 games.

iv. Calculate the probability that the World Series ends in 7 games.

b) Let X represent the number of games that the World Series will consist of this year. Compute the mean and variance of X.