The graph below shows the probability that a player who averages half a touchdown per game will score 0, 1, 2, 3, or 4 touchdowns in any given game:

How did I generate that? Using something called the Poisson Distribution, which Doug Drinen wrote about nine years ago. Let’s reproduce the relevant text here:

One simple model would be to note that this player should average .5 touchdowns per game and then view each game as a coin flip. Heads he scores in that game, tails he doesn’t.

That’s not a terrible model. It will give the guy 8 TDs per year in the long run. But it’s obviously lacking. In the long run, it will predict that half his games will be 1-TD games and the other half will be 0-TD games. Our years of experience reading box scores tell us that’s not realistic.

So why not break it down a bit further? Instead of viewing this guy as a .5-TDs-per-game player and then simulating 16 games each as a coin flip with probability .5, we could view him as a .25-TDs-per-half player and then simulate 32 halves each as a coin flip with probability .25. This idealized receiver will still average 8 TDs per year, but now he will have 2-TD games 12.5% of the time, 1-TD games 37.5% of the time, and 0-TD games half the time.

Better.

But why stop there? Let’s look at him as a .125 TDs-per-quarter player and simulate 64 quarters. I’ll spare you the calculations, but this would result in the following:

0-TD games:  58.62% of the time
1-TD games:  33.50% of the time
2-TD games:   7.18% of the time
3-TD games:   0.68% of the time
4-TD games:   0.02% of the time

Now that’s starting to look relatively realistic.

This “coin-flipping” model is called a binomial model, by the way. Let’s stop here and consider a couple of the assumptions implicit in the binomial model. In order to compute the above, we have assumed that each quarter (coin flip) is independent of the others. In other words, the above assumes that Chad Johnson‘s scoring in the first quarter tells us nothing one way or the other about whether he’ll score in the second quarter. There are all sorts of reasons why we might doubt that assumption. Scoring in the first quarter might be a clue that he’s playing against a weak secondary, which would indicate an increased chance of TDs in future quarters of the same game. On the other hand, scoring in the first quarter might cause the opposing defense to start double- or triple-covering him, thereby leading to a lower probability of future TDs. And that’s just the tip of the iceberg of possible ways this model fails to be literally correct.

But you know what they say: all models are imperfect, some are useful anyway. Let’s press on and see what happens.

What if we look at him as a .00833-TDs-per-minute player and then simulate 960 minutes each as a coin flip with probability .00833? What if we look at him as a .0001389-TDs-per-second player and then simulate 57600 seconds?

We’re getting into some obvious absurdity here, as this model would yield a chance of this receiver scoring a thousand (or much more) TDs in a season. It would be a very, very, very tiny chance — so tiny that for all practical purposes it could never happen — but a chance nonetheless. Furthermore, as we break the season down into more and more pieces, each of which is smaller and smaller, the calculations required are getting uglier and uglier.

Believe it or not, it turns out that the math can be simplified by breaking the season down into infinitely many pieces, each of infinitessimal length (technically, breaking it down into N pieces and then taking the limit as N goes to infinity). When you do that, what you get is this:

Prob. of having an N-touchdown game =~ e^(-1/2) (1/2)^n / n!

This is called a Poisson distribution with parameter 1/2 (the parameter 1/2 comes from the fact that our guy averages half a TD per game). When you plug that in for various values of n, you get this:

0-TD games:  60.65% of the time
1-TD games:  30.33% of the time
2-TD games:   7.58% of the time
3-TD games:   1.26% of the time
4-TD games:   0.16% of the time
5+-TD games:  0.02% of the time

Now, this is simply a mathematical model. Does it work in practice? Well, in 2014, Eric Decker joined the Jets, and Allen Hurns entered the NFL. Since then, both players have been roughly 0.5 touchdowns per game players: Decker had 11 touchdowns in 22 games with the Jets before scoring again in his 23rd game in week 10 against Buffalo, while Hurns had 12 TDs in 24 games prior to week 10, too (and he promptly scored once yesterday).

Decker has had exactly one touchdown in 8 of his last 9 games, while Hurns has had exactly 1 TD in now seven straight games. So that means they’re both kind of mucking up what we would expect, based on the Poisson Distribution. Over the last two years, Decker and Hurns have caught 0 TDs in a game 53% of the time, caught one touchdown 43% of the time, and 2 TDs in the remaining 4% of games. Where are the multi-TD games!!

But these are just two players. And frankly, expecting mult-touchdown games is probably expecting too much, although the current “quirk” of both players scoring exactly once nearly every game is fun nonetheless. What about over the broader trend? There have been 123 players in NFL history to play in 16 games in a season and score exactly 8 touchdowns. For what it’s worth, 19 of those players had zero multi-touchdown games, but that’s not exactly a surprising result. Overall, these players:

• Did not catch a touchdown in 59.7% of games (Poisson Distribution Estimate: 60.7%);
• Caught exactly one touchdown in 31.8% of games (PDE: 30.3%);
• Caught exactly two touchdowns in 7.6% of games (PDE: 7.6%(!));
• Caught exactly three touchdowns in 1.0% of games (PDE: 1.3%); and
• Caught exactly four touchdowns once, ^[1]This was by Tampa Bay tight end Jimmy Giles in 1985, who caught half of his touchdowns that year in a crazy loss in Miami. or in 0.1% of games (PDE: 0.2%).

That is pretty cool, I think.  The Poisson Distribution comes within 1.5% of projecting exactly how often an X-touchdown game will occur for every X.  In other words, Decker or Hurns should have a multi-touchdown game soon. ^[2]Just kidding! This would be the gambler’s fallacy. Math is fun, right?