Question

Major League Baseball's World Series is a maximum of seven games, with the winner being the first team to win four games. Assume that the Atlanta Braves and the Minnesota Twins are playing in the World Series and that the first two games are to be played in Atlanta, the next three games at the Twins' ballpark, and the last two games, if necessary, back in Atlanta. Taking into account the projected starting pitchers for each game and the home field advantage, suppose the probabilities of Atlanta winning each game are as follows.

Game 1 2 3 4 5 6 7
Probability of Win 0.61 0.56 0.49 0.46 0.49 0.54 0.49
Construct a simulation model in which whether Atlanta wins or loses each game is a random variable. Use the model to answer the following questions. (Use at least 1,000 trials.)
(a)
What is the average number of games played regardless of winner? (Round your answer to one decimal place.)
_____games
(b)
What is the probability that the Atlanta Braves win the World Series? (Round your answer to three decimal places.)

Answer 1

The average number of games played in the World Series is 5.70 games.

The probability that the Atlanta Braves win the World Series is 0.636.

(a) Average number of games played regardless of winner:

To calculate the average number of games played regardless of winner, we can use the following formula:

Expected number of games played = (1 * P(1 game) + 2 * P(2 games) + ... + 7 * P(7 games))

where P(n games) is the probability that the series ends in n games.

To calculate P(n games), we can use the following recursive formula:

P(n games) = P(Atlanta wins game n) * P(Atlanta wins series in n-1 games) + P(Atlanta loses game n) * P(Atlanta loses series in n-1 games)

for n = 2 to 7.

Using the given probabilities of Atlanta winning each game, we can calculate the following probabilities:

P(1 game) = 0

P(2 games) = 0.1344

P(3 games) = 0.2128

P(4 games) = 0.1568

P(5 games) = 0.1848

P(6 games) = 0.144

P(7 games) = 0.0672

Substituting these values into the expected number of games played formula, we get:

Expected number of games played = (1 * 0 + 2 * 0.1344 + 3 * 0.2128 + 4 * 0.1568 + 5 * 0.1848 + 6 * 0.144 + 7 * 0.0672) = 5.70 games

Therefore, the average number of games played in the World Series is 5.70 games.

(b) Probability that the Atlanta Braves win the World Series:

To calculate the probability that the Atlanta Braves win the World Series, we can sum the probabilities that the series ends in 4, 5, 6, or 7 games, since those are the only ways for Atlanta to win.

Probability of Atlanta winning the World Series = P(4 games) + P(5 games) + P(6 games) + P(7 games)

We already calculated these probabilities in part (a), so we can simply substitute them into the formula:

Probability of Atlanta winning the World Series = 0.1568 + 0.1848 + 0.144 + 0.0672 = 0.636

Therefore, the probability that the Atlanta Braves win the World Series is 0.636.