Ben sent me an interesting tweet over the weekend in reference to the Broncos-Chargers game:

Divisional home dogs 9-1 ATS on the season 🐶 https://t.co/uRzwe31ZYv

— Kelly (@kellyinvegas) November 1, 2020

An argument given for the Broncos (+3) to cover against the Chargers was that at this point in the year, home underdogs were 9-1 ATS in divisional games. The Broncos did end up covering, so this trend is now at 10-1 on the season.* Is this something to keep betting on as we move forward in the season, or is it just a massive coincidence?*

Just like in the previous Trend-Busters article, we will operate under the assumption that each game is a coin flip against the spread. Using the binomial distribution we can calculate the exact probability that that the home team covers 10 out of 11 divisional games as the underdog. This probability is roughly 0.005, or about 1 in 200. The R command *dbinom(10,11,0.5) *gives you this number if anyone wants to follow along.

** This seems like strong evidence that there is actually something to this trend**….

However, when looking at trends such as this, it is important to look at the quantifiers attached to the trend. Here, we have three: ** “home”, “underdog”, and “divisional”.** When calculating how likely a trend is, it is vitally important to see how many ways there are to change the quantifiers. If you have a set of quantifiers that can be arranged 200 ways, it should be EXPECTED that one of the arrangements attains that 1 in 200 probability. That’s more of an indication of searching for a set of quantifiers that fits a narrative as opposed to having a trend that has any amount of predictive power.

**How many ways are there to arrange our quantifiers?**

“Home” can be swapped with “away” and “underdog” can be swapped for “favorite”, giving us 4 total ways to arrange these two. ** Is the narrative for home underdogs covering a disproportionate amount of divisional games any more or less convincing than the one for home favorites, away favorites, or away underdogs? I don’t think so.** If we were talking about winning outright, I could see an argument for home underdogs playing with a chip on their shoulder, but no team goes into a game saying “Vegas thinks we are going to lose by 6. Let’s make sure we lose by at most 4!” Accounting for this already drops our 1 in 200 trend to a 1 in 50 trend.

The “divisional” quantifier is much tougher. For one, divisional games add a rivalry aspect which definitely does have some merit as having predictive power for keeping games close. The other thing that makes this tough is figuring out exactly how many ways there are to change the “divisional” quantifier. I’m going to use one interpretation, but there are definitely other possible interpretations that will change the answer. Let’s start by defining what a division is in the NFL. There are 8 total divisions and each is a collection of 4 teams, with no overlap in the teams between the divisions.

One way to look at the divisional quantifier is to say that the divisions we actually have should be equally likely as any other possible way to arrange the divisions, like some alternate universe where the “AFC West” contains the Broncos, Lions, Jets, and Eagles. To do this, we calculate how many ways there are to split 32 teams into 8 divisions of 4 teams. One other note is that I’m doing this so that the division labels themselves don’t matter, so for example if we shuffled all of the teams and the only thing that changed was that all of the NFC East teams swapped with all of the NFC West teams, that shuffle would be the same as our real life divisions from the perspective of what games are “divisional games”.

**Heavy math warning:**

First, we have to calculate the total number of ways to arrange all 32 teams without regard to any kind of divisional split. This is an example of a permutation. The number of ways to permute 32 teams is 32 factorial, written 32!. This is equal to 263,130,836,933,693,530,167,218,012,160,000,000. Thats…a big number.

Once we have a permutation, how do we turn it into divisions? One way would be to take the first 4 teams in the permutation and make that “division 1”, the second 4 teams “division 2” and so on. Another way would be to count all 32 teams off 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, etc. Basically, we can arrange the numbers 1,1,1,1,2,2,2,2,3,3,3,3,…,8,8,8,8 any way we want, and assign the teams with a 1 to division 1 and so on. How many ways are there to do this? This is an application of the multinomial theorem, and applying it to our problem, we get that there are exactly 110075314176 ways to assign divisions to a particular permutation. Dividing the total number of permutations by the number of possible division-assignments gives us the total number of ways to assign 32 teams to 8 divisions of 4 teams, which turns out to still be massive (about 10^24).

There is still one more step, which is to remove the division labels (e.g. swapping “NFC East” for “AFC East”). To do this, we further divide by the total number of ways to permute the 8 division labels, which is 8!, or 40320. Dividing this out gives us exactly 59,287,247,761,257,140,625 ways to assign 32 teams to 8 equally sized divisions when we don’t care about division labels.

**What does this mean for the trend?**

It means that even though in our universe where the NFL divisions are what we know them to be, home underdogs being 10-1 in divisional games is only a 1 in 200 event, the fact that our set of “divisions” is only 1 out of more than 59 QUINTILLION ways to make “divisions”, it’s not absurd at all that this 1 in 200 event has happened. We should expect it to happen in 1 out of 200 of these “divisions”, and we just happen to be living in one where it did.

**Added from Ben-**

Great stuff, Nick. The whole idea here is that this trend is one not to chase. At this rate, divisional dogs would go 20-2 ATS on the season. Do we really think that is going to occur especially after the aforementioned math behind this? Trends are fun to explain but you should be careful betting on trends. Did the Broncos beat the Chargers mainly because they are both in the same division, or were there more prominent factors that caused the Broncos to win this one such as the altitude factor (Chargers looked bad in the second half), or Lance Lynn blowing his third game up 2 scores in the 4th quarter. Or the Broncos giving the ball to Phillip Lindsay, Or Joey Bosa getting hurt in the second half. Stay tuned for more trend busters.