This year (2005), the baseball teams qualifying for the playoffs in the American League will be determined by two 3game 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 9465 record while Boston and Cleveland both have a 9366 record. Depending on the outcomes of the two 3game series, the team to be eliminated (that is, the only team among those three not going to the playoffs) will be:
BOSNYY 30 
BOSNYY 21 
BOSNYY 12 
BOSNYY 03 

CHICLE 30  CLE  CLE  CLE  BOS/CLE 
CHICLE 21  CLE/NYY  CLE  BOS/CLE  BOS 
CHICLE 12  NYY  BO/NY//CL  BOS  BOS 
CHICLE 03  NYY  BOS/NYY  BOS  BOS 
Here, for example, BOS/CLE means that there will be an additional tiebreaking game to decide whether Boston or Cleveland is eliminated. The notation BO/NY//CL means that there will be two tiebreakers: 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