Wild Card Calculations
As explained elsewhere on
the site, all calculations are based on the assumption
that every team has equal ability. Consequently, each game is treated as a
50-50 matchup, with either team having an equal chance of winning.
When division winners
automatically qualify for the playoffs and additional teams qualify as wild
cards, the probability of making the playoffs for a given season is the sum of:
- the probability of winning the division, and
- the probability of earning a wild card berth.
When every division within a
conference or league contains the same number of teams, these probabilities are
relatively straightforward to calculate.
The probability of winning
the division is simply:
P(win division) = 1 / D
where D is the number of
teams in the division.
The probability of earning a
wild card berth is:
P(wild card) = (1 − (1 / D)) × (W /
(C − d))
where:
- D = number of teams in the division
- W = number of wild card teams in the conference
or league
- C = number of teams in the conference or league
- d = number of divisions
This formula works because a
team must first fail to win its division and then earn
one of the available wild card spots among the remaining teams.
When Division Sizes Differ
The calculation becomes more
complicated when divisions are not all the same size.
To illustrate why, consider
an extreme example with three divisions:
- Division A: 2 teams
- Division B: 6 teams
- Division C: 10 teams
The probability of winning
the division is still easy to calculate:
- Division A: 1/2
- Division B: 1/6
- Division C: 1/10
However, the probability of
earning a wild card berth is no longer the same for every team.
Assume there is only one
wild card berth available. A team must finish second in its division to have
any chance of earning that berth. Finishing first already qualifies the team
for the playoffs, while finishing third or lower guarantees that at least one team
in the same division finished ahead of them.
The key point is that the
number of wins typically required to finish second depends heavily on the size
of the division.
In Division A, the
second-place team could be one of the worst teams in history and still finish
second simply because there are only two teams. As a result, the distribution
of wins for the second-place team is very wide, and its average number of wins is
relatively low.
In contrast, the second-place
team in Division C must outperform eight other teams while still finishing
behind the division winner. Consequently, the distribution of wins is much
tighter and centered at a much higher value.
Using order statistics, the
expected number of wins for the second-place team in a 162-game baseball season
is:
|
Division |
#
Teams |
Avg
Wins |
90%
of Seasons Range * |
|
A |
2
teams |
77.4 |
69
- 86 |
|
B |
6
teams |
85.1 |
80
– 91 |
|
C |
10
teams |
87.4 |
83
- 93 |
* 90% of the time, the number
of wins would fall in this range
These differences are
substantial. A second-place team from a large division is much more likely to
have a record strong enough to earn a wild card berth than a second-place team
from a small division.
Although real life leagues
typically differ by only one or two teams per division rather than the extreme
example above, the effect is still large enough to influence the probabilities.
Therefore, whenever wild card teams are included, the playoff probabilities on this
website account for the actual division sizes as well as the overall conference
or league structure.
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