Heading to the Playoffs

October 5, 2015

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?


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)


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.