So, was it the right call to go for it on 4th and goal?

Guess what, I'm going to use non-intuitive math to come up with an unexpected conclusion.  You other stats heads are welcome to check my work, since I've had beer and wine and am a bit fuzzy right now.  We're going to use Brian Burke's WPA and EPA points to try and figure this out.  Details and conclusion after the jump.

First, we have to check the numbers.  Burke's Win Probability Calculator is pretty cool.  It factors in game situations from every nfl game for several years to come up with odds of winning, and expected points, from every possible game situation.  So let's look at our ingredients.

Denver has 4th and goal from the 1 yard line with 13:33 left in the 4th quarter, up by 4 points.  Our WPA here was 0.81.

If we had scored the touchdown, our WPA after a touchback would have been 0.91.

If we had kicked the field goal, our WPA after a touchback would have been 0.82.

Failing the fourth down (or missing the field goal) gave us a WPA of 0.70.  Tennessee at their 1-yard-line had an EPA of -0.53.  This is because when backed up that far, the odds were still that we would have scored points before they would have.

To other things we know - touchbacks *do* have value to returning teams.  A touchdown tends to be worth 6.3 EPA, and a field goal tends to be worth 2.3 EPA.

So let's plug in the numbers.  You can gloss over this point if you want.  We're going to look at it in terms of EPA, and then WPA.

EPA:

Let's say that 19-yard field goals are made 95% of the time.  We can quibble but I think that's a safe assumption.  We have a 95% chance of getting the EPA of a made field goal (2.3), and a 5% chance of getting the EPA of a missed field goal (0.53, since Tennessee is backed up so much).  So the total EPA of a field goal *attempt* is:

(0.95 * 2.3) + ((1-0.95) * 0.53) = 2.21 EP

Ok, now let's look at the touchdown.  We don't know how likely the touchdown is, but we want it to be worthwhile compared to the field goal attempt.  In other words, how likely should our touchdown-making-chances be to go for the TD instead of the field goal?  Here, we solve for that variable, using the numbers of the touchdown EPA (6.3) and the failed touchdown EPA (0.53, again since Tennessee is so backed up).

(x * 6.3) + ((1-x) * 0.53) = 2.21 EP

6.3x + 0.53 - 0.53x = 2.21 EP

5.77x = 1.68 EP

x = 29.12

How do we read this?  It means that if we feel we had a 29% chance (or better) of scoring the touchdown, then it was the right call to go for it.

HOWEVER...

Hold your Broncos here.  There's another way to look at it.  Let's do the same analysis using WPA.

If we make the field goal, we have a 95% chance of our WPA being 0.82, and a 5% chance of our WPA being 0.70.  So what is the average WPA of our field goal attempt?

(0.95 * 0.82) + (0.05 * 0.70) = 0.814 WPA on average.

So then what about the touchdown with WPA?  What likelihood of touchdown conversion do we need to have in order to beat that average WPA?  Here we're comparing our 0.91 WPA of a made touchdown, to the 0.70 WPA of the missed touchdown.

(x * 0.91) + ((1-x) * 0.70) = 0.814 WPA

0.91x + ((1-x)(0.70) = 0.814

0.91x + (0.7-0.7x)  = 0.814
0.21x + 0.7 = 0.814
0.21x = 0.114
x = 54.29

How do we read this?  It means that if we feel we had a 54% chance (or better) of scoring the touchdown, then it was the right call to go for it.

WHAT??

Why are those numbers so different?  The answer lies in the difference between EPA and WPA.  EPA only factors in likely points given a field position.  It doesn't take game factors into account - time left, score margin.  WPA takes remaining time and score margin into account.

In other words, EPA is saying that in general, going for it in that situation is the right decision if we feel we have a 29% chance of converting.

However, given that particular game situation - up by four points in the fourth quarter - WPA is telling us that the field goal was worth more than it would have been in the average game situation.  When factoring in remaining time and score differential, we should only have gone for it if we felt we had a 54% chance (or better) of converting.

CONCLUSION

Honestly, I think we probably had a 29% chance of converting.  But given our offensive run blocking and Tennessee's DL, I don't think we had a 54% chance of converting that 4th down.  I think Fox was right only if you ignore the game situation we were in.  I think he was right theoretically but not practically.  In short, I think Fox screwed up.  We should have kicked the field goal, and if Fox didn't realize it during the game, he should be realizing by now it was a mistake.  Given his post-game comments though, it doesn't look like he sees it that way.