Problem: Suppose that Boston Red Sox and New York Yankees (two American League baseball teams) are scheduled to play a best of three series. The winner of the series will be the first team that wins two of the three games. The probability that the Red Sox win a game in their home stadium is 0.6 and the probability that Yankees win their home game is 0.57. Next, suppose that you place a bet on each game played where you win $500 if the Red Sox win and you lose $520 if the Red Sox lose the game. In parts 1-3 below, assume that the outcomes of the games are independent of each other.

Part 1: If the first game is played in Boston, the second game is played in New York, and the third game (if it becomes necessary) is in Boston, then complete parts (i)-(v) below. (i) Calculate the probability that the Red Sox will win the series. (ii) Construct a probability distribution for your net win (X) in the series. Calculate your expected net win (the mean of X) and the standard deviation of X. (iii) Use Excel or R to create 10,000 random values for X. Let these random values be denoted by Y. Use these Y values to estimate your expected net win by using a 95% confidence interval. Does this confidence interval contain E(X)? (iv) Construct a frequency distribution for Y. Next, use the Chi-squared goodness of fit test to verify how closely the distribution of Y has estimated the distribution of X. (v) Use your observations of parts (ii) and (iii) above to describe whether your betting strategy is favorable to you. Write a summary of your observations and analyses in the Word document.

Part 2: Repeat part 1 above but assume that the first game is played in New York, the second game is played in Boston, and the third game (if it becomes necessary) is in New York.

Part 3: Repeat part 1 above but now assume that the series is a best-of-five series where the first team that wins three games wins the series with games alternating between Boston and New York, with the first game being played in Boston.