Joe Morgan made the following statement on ESPN tonight:
The NL West is the only division that hasn't won the World Series in the past seven years. Now, that's amazing.
Except, as my brother points out (drawing on his years of accumulated baseball wisdom), there are six divisions.
Start with the assumption that each division is equally skilled, and winners are determined purely by chance. In this scenario, each division is expected to win 1.167 World Series (7/6), which in real terms translates into one division winning the Series twice and the other five divisions each winning once.
But let's gets more specific. If each division is equally likely to win, then what is the probability that one division has no wins in seven years? The probability of any division winning in a single year is 1/6, which means the probability of any division losing in a single year is 5/6. The probability of losing for seven years is which equals 27.9%. So, even if every division were equally matched, the chance that one division would have no championships in 7 years is almost one in three.
You might argue this this ignores the wild card! Each division has an extra chance to make the playoffs - so there are 8 teams competing for the title, two divisions in opposing leagues having sent two teams each. But if each division is equally matched, then the wild card is not an advantage unique to any division, and so it does not affect the probability of any individual division winning ex ante. Of course, once it is known whether a division holds the wild card spot (and consequently is sending two teams to the playoffs), then the advantage turns to that division. There have been enough wild card World Series winners (though the total sample is small since the format was introduced) that the effect - conditional on winning the wild card, which is not part of this question - is certainly non-negligible.
We can be more advanced, however, in dealing with the wild card situation. Each division has a different number of teams, so assuming that each division is equally skilled is not the same as assuming each team is equally skilled. If each team is equivalent, the NL West sends one team to the playoffs by definition and has a 4/13 chance of additionally sending a wild card team. 4/13 arises from there being 16 teams in the NL and 5 teams in the NL West. The wild card is fought for among teams that don't win the division, which implies 4 teams from the NL West out of a total of 13 non-winners.
Once an evenly-matched team makes the playoffs, their probability of winning the World Series is or 12.5% (3 rounds, 50% of winning each one). The NL West has two ways to make the playoffs: a division winner with probability 1 and a wild card team with probability 4/13. Thus, the combined probability of the NL West winning the World Series is or about 16.3%.
This is slightly less than the naive probability of 1/6, or 16.67%, calculated above because the NL West has one less team than the NL Central, which puts it (and the similarly-numbered NL East) at a slight disadvantage in the Wild Card. We can continue as before, using the formula , which yields a 28.7% chance of the NL West not winning a single World Series in 7 years. As expected, the number is slightly higher than the 27.9% calculated earlier, but the interpretation is the same: if every team is equally likely to beat every other team, there is almost a 1 in 3 chance the NL West won't win a championship in 7 years.
And it's not for a lack of trying - an NL West team has actually been in the World Series twice in the last 7 years - and that's right in line with what you'd expect, given 7 berths available to 3 divisions. That they haven't won either time, however, is just a 1 in 4 event.
But Tampa Bay making the World Series? Now, that's amazing.