Question
In Major League Baseball, the National League has three divisions, the East and Central divisions with 5 teams each, and the West division with 4 teams. Four teams in the National League will make the playoffs, the three division leaders, and a wild card team. The wild card team is the team with the best record in the league, not including the three division leaders. Assuming all teams are equally good, what is the probability of a team in each division making the playoffs? Assume any ties are broken randomly.
Solution
The difficult part of this question is determing the probability that the Wild Card team will fall in any given division. Let's consider the American League East first. There are six ways the wild card team could be in the American League East, as follows.
American League East has both first and second place teams in the entire American League. The probability of this is (5/14)×(4/13) = 10.9890%.
American League East has both first and third place teams, but not the second, in the entire American League. The probability of this is (5/14)×(4/13)×(9/12) = 8.2418%.
American League East has both second and third place teams, but not the first, in the entire American League. The probability of this is (5/14)×(4/13)×(9/12) = 8.2418%.
American League East has both first and fourth place teams, in the entire American League, and the second and third place teams fall one each into the other two divisions. The probability of this is = (5/14)×(4/13)×((5/12)×(4/11)+(4/12)×(5/11)) = 3.3300%.
American League East has both second and fourth place teams, in the entire American League, and the first and third place teams fall one each into the other two divisions. The probability of this is = (5/14)×(4/13)×((5/12)×(4/11)+(4/12)×(5/11)) = 3.3300%.
American League East has both third and fourth place teams, in the entire American League, and the first and second place teams fall one each into the other two divisions. The probability of this is = (5/14)×(4/13)×((5/12)×(4/11)+(4/12)×(5/11)) = 3.3300%.
Add these probabilties together and you get 37.4625%. It is obvious the probability should be the same for the American League Central. The probability for the American League West thus has a probability of 100% - 2×37.4625% = 25.074925% of having the wild card team.
The probability any given team will be a division leader is 20% in the East and Central divisions, and 25% in the West division. These are also the probabilities for the wild card spot, given that it is in the same division. So, the probability that a team from the East or Central divisions makes the playoffs is 20% + 37.4625%×20% = 27.492500%. The probability that a team from the West division makes the playoffs is 25% + 25.074925%×25% = 31.268731%.
Extra Credit
The probability the National League East or West division will get the wild card is 29.670330% each, and 40.659341% for the Central division. The probability any given team in the East or West will make the playoffs is 25.934066%, and 23.443223% for Central.
Summary
American East 27.492507%
American Central 27.492507%
American West 31.268731%
National East 25.934066%
National Central 23.443223%
National West 25.934066%