# The Reverse Sweep

If you’ve been following the NBA playoffs, you will know that we are on the verge of a potentially historic event: the first ever reverse-sweep! The Boston Celtics will face the Miami Heat at Boston on Monday May 29th in game 7 of the Eastern Conference finals to determine which team will advance and face the Denver Nuggets in the NBA finals. The Heat won the first 3 games of the series, and then the Celtics won the following 3 games. If the Celtics win on Monday night, they will complete a “reverse sweep”. This has never happened in any best-of-seven series in NBA playoff history! But how rare is a reverse sweep, actually?

I compiled some statistics for the NHL (hockey), NBA (basketball), and MLB (baseball). Each of these sports has best-of-seven series for at least a portion of their playoffs. Playoff formats have changed throughout the years, so I only counted playoff series that used a best-of-seven format. I also excluded any series where games ended in a tie (this happens in baseball). Finally, I only considered NHL, NBA, and MLB, but there are other leagues that also use a best-of-seven format. Here was the breakdown:

 Best-of-7 series Started 3-0 Lasted 7 games Reverse sweeps NHL hockey 755 205 9 4 NBA basketball 601 150 3 0 MLB baseball 184 38 2 1

The numbers are so low for baseball because the MLB only uses best-of-seven series for the world series, ALCS, and NLCS, whereas hockey and basketball use many more. Here is a more detailed breakdown of how the best-of-seven series that started 3-0 finished up.

But is this in-line with what we might expect to see? Let’s find out…

## Simple model 1: home-court advantage

The distribution of outcomes depends on many factors, including home-field/court advantage, player injuries, coaching decisions, and so forth. Moreover, the data are limited, so it’s impossible to truly know what’s going on. So we will do the next best thing: see what we would expect to have happen under different sets of reasonable assumptions.

We will start with home-court advantage. Turning our attention to basketball specifically, it is well-documented that the “home team” has a significant advantage. In particular, the home team wins roughly 60% of the time, on average, during the regular season. During the playoffs, this number figure increases to 65%. In the NBA, the higher-seeded team gets to play the first two games at home, then then next two games away, then it alternates from that point on. So the format is: H,H,A,A,H,A,H.

Let’s assume that home-court advantage is the only factor that determines the chance of winning. Suppose the probability that for every game, the home team wins with probability $p$. Then:

• The higher-seeded team will go up 3-0 with probability $p^2(1-p)$.
• The lower-seeded team will go up 3-0 with probability $p(1-p)^2$.
In this fashion, we can calculate: “given that the series starts 3-0, the distribution of ultimate outcomes is…”. This leads to the following distributions for different values of $p$:

This doesn’t seem to quite match the data, so even though we have aggregate data about what happens on average when it comes to home-court advantage, series that start 3-0 are evidently not typical! Let’s try something else…

## Simple model 2: one team is just better

Another possibility is that one team is simply better than the other. In this model, we will assume that one team wins with probability $p \gt 0.5$, regardless of home-court advantage. In this scenario:

• The better team goes up 3-0 with probability $p^3$.
• The worse team goes up 3-0 with probability $(1-p)^3$.
In this fashion, we can again calculate: “given that the series starts 3-0, the distribution of ultimate outcomes is…”. This leads to the following distribution for different values of $p$:

This seems to better align with the distributions we saw above for the historical outcomes in the NHL, NBA, and MLB. But then again, these are just simple models, so one shouldn’t read too deeply into them!

## Who will win game 7?

Of course, there is no way to know. However, I don’t think there is any reason to believe the NBA is somehow different than other professional sports with regards to the likelihood of reverse sweeps. Ultimately, if two teams make it to game 7 in a best-of-seven series, I think it’s fair to say that they are evenly matched. For the NHL, this happened 9 times, and the outcome was 5-4 (i.e., there was a reverse sweep 4 of 9 times). For MLB, this happened 2 times, and the outcome was 1-1. For the NBA, this happened 3 times, and the outcome was 3-0. If you flip a fair coin 3 times, there is a 1/4 chance that it will land the same all 3 times. So what we’ve seen is not that unlikely, and it’s surely just a matter of time before a reverse sweep is achieved in the NBA.

That being said, the NBA will witness it’s 4th chance at a reverse sweep on Monday May 29th. I, for one, hope we get to see history happen. Go Celtics!