Math 425: Another (time-consuming) practice problem for the first midterm exam

This year (2005), the baseball teams qualifying for the playoffs in the American League will be determined by two 3-game series, to be played on September 30 - October 2, 2005, between Boston Red Sox -- New York Yankees and Chicago White Sox - Cleveland Indians, respectively. The Chicago team has already qualified for the playoffs, along with Los Angeles Angels. Each of the other three teams mentioned above (Boston, Cleveland, and New York) has a chance of qualifying. In fact, exactly two of these three teams will go to the playoffs. As of now (September 30, 7 PM EST), New York has a 94-65 record while Boston and Cleveland both have a 93-66 record. Depending on the outcomes of the two 3-game series, the team to be eliminated (that is, the only team among those three not going to the playoffs) will be:

 BOS-NYY3-0 BOS-NYY2-1 BOS-NYY1-2 BOS-NYY0-3 CHI-CLE3-0 CLE CLE CLE BOS/CLE CHI-CLE2-1 CLE/NYY CLE BOS/CLE BOS CHI-CLE1-2 NYY BO/NY//CL BOS BOS CHI-CLE0-3 NYY BOS/NYY BOS BOS

Here, for example, BOS/CLE means that there will be an additional tie-breaking game to decide whether Boston or Cleveland is eliminated. The notation BO/NY//CL means that there will be two tie-breakers: first, between Boston and New York; then, between the loser and Cleveland. In this scenario, the loser of the second game is eliminated.

Assuming that each game is won by one of the two teams with probability 1/2 regardless of the outcomes of other games, determine the probabilities that Boston (resp., Cleveland, New York) are eliminated.

A justification for the table (not needed to solve the problem) can be found here.

Sergey Fomin, September 2005