# What Are The Chances Of One City Winning At Least Three Out of Four Titles In The Same Year?

With the Boston Bruins set to take on the St. Louis Blues for the Stanley Cup Finals tonight, Boston is on the verge of being the first city since Detroit in 1935 to hold at least three of the four major North American professional sports titles (sorry, MLS). Earlier this postseason, I wrote an article detailing Bostonâ€™s chances of achieving the â€śBoston Slamâ€ť and holding all four titles at the same time, and then detailed Bostonâ€™s chances on the HSAC Twitter page throughout the two conference semifinal series. Unfortunately, the chances never got much higher than 1% as the Celtics flamed out in the Eastern Conference semifinals to the Milwaukee Bucks in 5 games.

However, the Bruins have just kept on winning, winning their last seven games and completing an impressive sweep over the Carolina Hurricanes in the Eastern Conference Finals to claim the Prince of Wales Trophy. During this excruciatingly long time period between the Conference Finals and the Stanley Cup (11 days for the Bruins, 6 for the St. Louis Blues), I was asked on Twitter about what the chances of one city winning three out of the four major titles in the same year.

I immediately realized that this was an interesting combinatorics problem that did not require super sophisticated math, and decided to set out on finding an answer.

There were some interesting things to think about with this problem. The first is that there are effectively five ways for a city to hold three out of four championships. They can win all four in the same year, or they can win exactly three out of four, leaving one sport out (four combinations). The second is that some cities have teams in exactly three out of the four major sports (like Atlanta, Pittsburgh or Houston) and thus can only have at least three out of four if all of their teams win. Meanwhile, some cities have exactly one team in all four sports (like Boston, Philadelphia or Detroit) and others have more than one team in at least one sport (like New York, Los Angeles or Chicago). Finally, some cities (like Seattle, St. Louis or Baltimore) do not have teams in at least three sports and thus are ineligible from achieving this feat. Thus, each city does not have an equal chance of attaining this feat. Finally, it is impossible for more than one city to achieve this in the same year (as that would mean at least six championships!), so it is sufficient to independently sum each individual cityâ€™s probability of winning at least three out of four.

In order to compute each cityâ€™s chances of winning at least three out of four titles, I made a simplifying assumption that in a given league, each teamâ€™s probability of winning a championship is uniform. Thus, it is assumed that the Patriots have a 1/32 chance of winning the Super Bowl, while the New York Yankees have a 1/30 chance of winning the World Series. This is not a totally reasonable assumption, as it ignores the effect of teams in â€śbig marketsâ€ť having the ability to spend more money and thus will have a higher probability of winning a championship than a team in a smaller market. However, in absence of any formal modeling of this effect,, a uniform distribution will have to do.

I wrote the following function in R to determine the chances of a city winning at least three out of four, given the number of teams that city has in each individual league, based on the five potential combinations described above.

For example, to compute the probability of New York doing this, you would feed in (3,2,2,2) to the function because New York has 3 NHL teams (yes, the Devils count), 2 NBA teams (yes, the Knicks also count even though New Yorkers would prefer if they didnâ€™t exist), 2 NFL teams (technically, although they both play in New Jersey too) and 2 MLB teams. For Boston, you would feed in (1,1,1,1) and for Atlanta you would feed in (0,1,1,1) since Atlanta does not have an NHL team (#RIPThrashers).

In this analysis, we used 20 cities that had a team in at least three of the four major professional sports leagues. Some subjective judgements were made in terms of determining what counted and what didnâ€™t. For example, I decided that the Green Bay Packers were a Milwaukee team despite being a two hour drive away and I also decided that all teams in the San Francisco Bay Area were considered to be from the same city, thus grouping together San Francisco, Oakland and San Jose. On the flip side, I decided that teams from Nashville/Memphis and Charlotte/Raleigh should not be combined, so none of the teams from those four cities were considered in this analysis.

After computing the above function for each city, I found each individual cityâ€™s probabilities of winning at least three out of four.

When we sum up the individual probabilities, we get 0.43%. Thus, ignoring the â€śbig marketâ€ť effect and assuming all franchises in a given league have a uniform probability of winning the championship, we would expect one city to win at least three out of four titles about once every 227 years.

As a note, these calculations will be slightly altered when the new Seattle NHL team enters the league in 2021/22, giving Seattle 3 teams and adding to the denominator for the NHL calculations.

It is also interesting to calculate the probability of Detroit achieving the same feat in 1935. Back then, there was no NBA, the MLB had 16 teams, the NHL had 8 (as the NY Americans and Montreal Maroons had yet to fold to give the traditional Original Six), and the NFL (in the pre Super Bowl era) had 9. Four cities (New York, Boston, Chicago and Detroit) had at least one team in all three leagues. New York was boosted by 3 MLB teams (New York Giants and Brooklyn Dodgers), 2 NFL teams (New York Giants and Brooklyn Dodgers) and 2 NHL teams (Rangers and Americans), while Chicago was boosted by having two teams in both the NFL (Bears and Cardinals) and MLB (Cubs and White Sox) and Boston had 2 MLB teams (Red Sox and Braves). Thus, Detroitâ€™s probability of winning all three was 0.08%, Bostonâ€™s was 0.17%, Chicagoâ€™s was 0.35% and New Yorkâ€™s was 1.04%. If you sum all four of those up, the probability of a city winning three championships in 1935 was 1.65%, and we would expect it to happen once every 60 years given the league compositions in 1935.

Another interesting thing to study with this is when did the probability of a city winning at least three out of four peak, and how high was this probability? This was likely in 1947/48, during the 2nd year of the NBA.The NBA had 8 teams, the NHL 6, the MLB 16 and the NFL 10. There were six cities that had the chance to win three titles, and Chicago (boosted by having two teams in both the MLB and NFL) had the highest probability with 1.25%. The overall probability of one city winning at least three out of four was 3.37%, meaning we would expect this to happen about once every 29 years.

It would be interesting to control for the big market effect to redo these calculations. If you have any ideas for how to go about this, or have any questions/comments about the article, please feel free to reach out to me on Twitter @andrew_puopolo.