Why the world will not end if Cubs win the 2015 World Series


The Chicago Cubs are currently (as of October 20 2015) playing the New York Mets in the 2015 National League Championship Series, potentially eight victories away from winning the World Series, something that has not happened since 1907. As baseball lore goes, the team is cursed, and a World Series victory would be an event of cosmic proportions, possibly signaling end of times itself. This begs the question: just how unlikely is an 107-year drought?
     For simplicity, let us first assume that in any given year, each team has equal chance of winning the WS. From the Cubs' point of view, the probability of it not winning the WS since 1907 is about 0.38%, i.e. pretty unlikely. However, what is the probability that some (at least one) team hasn't won the WS since 1907? By my simulation*, it is about 6%, i.e. not probable, but not impossible. This is similar to the Birthday Paradox: if you walk into a room of 30 people, the probability that someone has the same birthday as you is low (~7.6%), but the probability that some pair of people share the same birthday is quite high (~70%).
     While we are on the topic, the probability that at least 2 teams did not win the WS from 1919 to 2003, as was the case with the Cubs and the Red Sox, is a little over 1%. Not something anyone would have predicted in 1918, but had the two met in the 2003 WS (both teams advanced to their respective league's Championship Series that year), the world likely would not have ended either.
     We should remember though that while the Cubs and the Red Sox received all the attention, there was a third team that did not win a championship between 1919 and 2003: the Chicago White Sox, who, before winning the WS in 2005, last won in 1917. Together this trio of futility does push the limits of incredulity, as the probability of at least three teams concurrently trapped in 85-year droughts appears to be less than 0.04%.
     By the way, as a die-hard Cleveland Indians fan, what is the probability that, conditioning on the Cubs's drought, some other team hasn't won the WS since 1948, as is the case with the Indians? It is actually about 53%! So sure, the Indians have been very unlucky, but it's almost as if a flip of a coin decided that some team has to be that unlucky, and that team just happens to be the Indians (well, that, and decades of mismanagement).
     Before we leave this discussion, I want to address what happens if, as it happens in reality, not all teams have equal chance of winning the WS. Major League Baseball was originally comprised of 16 teams, and more teams were gradually added starting in 1961 to make up today's 30. It is fairly clear that the probability of an 107-year drought decreases if the expansion teams have had less than uniform odds of winning the WS. (This is a reasonable assumption, as a new franchise needs time to build a contending team and a fanbase; I counted only 9 of the 54 WS since 1960 being won by expansion teams.) On the other hand, if we fix the winning odds of the expansion teams, then I believe that the probability of drought is minimized if the 16 original teams are always equally likely to win the WS. I do not have a rigorous proof for this, but the extreme example where one team always has zero chance of winning suggests that the probability of drought increases as distribution on winning shifts away from uniformity (over the original teams). Thus assigning zero chance of winning to the expansion teams and 1/16 chance of winning to the original teams every year yields a lower bound on the probability in question. Simulation puts the probability of at least one 107-year drought at about 1.5% in this model.
     In conclusion, if 2015 is the year that the Cubs's drought finally ends, it would be the end of a streak that, while surely painful for generations of Cubs fans, is not really all that mathematically remarkable in and of itself. However, do notice the droughts of the Red Sox (no WS from 1919 to 2003, winning in 2004) and the White Sox (no WS from 1918 to 2004, winning in 2005), and how neatly the Cubs winning the 2015 WS would completely the symmetry. As we speak, the Cubs are down 3 games to none to the Mets. If they still manage to make it to the World Series and win, well, that may indeed be of some numerological interest.

*One might ask why, if we know $p$, one team's probability of not winning the WS since 1907, we cannot simply compute the probability of some team not winning the WS since 1907 with the formula $1-(1-p)^{\text{numTeams}}$. The reason is that teams's records are not independent. To be fair, the dependence is very weak and the aforementioned formula does yield about 6% as well.