Problem of the Week Archive

Heading to the Playoffs – October 5, 2015

Problems & Solutions The regular season of major league baseball ended on Sunday, October 4th. Each team plays 162 games throughout the season, which began on April 5th. With only five days left, the race for the play-offs was still going in the American League West.

As of September 30th, the Texas Rangers were leading their division with a record of 86 wins and 72 losses while the Houston Astros were in second with a record of 84 wins and 75 losses. At that time, Rangers fans were trying to figure out how close they were to clinching a division title. This statistic is called the “Magic Number” for the first place team and the “Elimination Number” for the second place team.

The Magic Number is determined by adding the leading team’s number of wins to the number of losses for the trailing team, and subtracting that total from 163. What was the Magic Number for the Rangers?

To calculate what the Magic Number for the Rangers was on September 30th, we need to add the number of wins the Rangers had to the number of losses the Astros had and subtract the sum from 163. The Magic Number for the Rangers was: 163 – (86 + 75) = 163 – 161 = 2.

As of Wednesday, September 30th, the Astros had three games remaining in their regular season and the Rangers had four games remaining. In order for the Astros to finish with more total wins than the Rangers by the end of the regular season, what will each team’s win-loss record need to be for their remaining games? (Note: the Astros could win the division by forcing a tiebreaker, but this would be in addition to the regular season and should not be considered for this problem)

The Astros were trailing the Rangers by 86 – 84 = 2 games. In order to have more wins than the Rangers, the Astros would have needed to win a minimum of 3 games, bringing their total wins to 87. Since they only had 3 games remaining, the Astros needed to go 3-0, 3 wins and 0 losses, in order to have a chance of beating the Rangers. If the Astros won the 3 necessary games to bring their total to 87 wins, then they would need to Rangers to lose all of their remaining games, since even one win by the Rangers would bring their total to 87 wins, a number the Astros are unable to beat. Since the Rangers had four games remaining they would have had to go 0-4, 0 wins and 4 losses. The only senario in which the Astro would finish the regular season with more total wins than the Rangers would be for their records from September 30th to October 4th to be 0-4 for the Rangers and 3-0 for the Astros.

Based on the previous problem’s answer, what is the probability that the Astros win the division by the end of the regular season? Assume that there is an equal probability of a win or loss for each game played. Express your answer as a common fraction.

Since we assumed there was an equal probability of a win or loss for each game, the probability of the desired outcome for each individual game is ½. Between the Rangers and the Astros, there are seven games to be played and only one outcome results in the Astros winning the division. The probability that the Astros win three out of three games is ½ × ½ × ½ = 1/8. The probability that the Rangers lose four out of four games is ½ × ½ × ½ × ½ = 1/16. The probability of both happening, and the Astros winning the division by the end of the regular season, is 1/8 × 1/16 = 1/128.