First Down Probability

For every series beginning with a 1st down and 10 in the NFL, an offense achieves another 1st down (including scoring a TD) 66% of the time. So if an offense is to be graded on a 1st down play, it should at least gain enough yards so that they have at least the same probability of sustaining the drive or scoring. In this article, I’ll begin a discussion of utility in football by examining the break even point in a series of downs. In other words, what would you rather have, a 1st and 10 or 2nd and 7? What about 2nd and 6? 5?

When any team has the ball, it's typically trying to sustain a drive. It would be nice if they threw a 50 yard bomb for a touchdown, but usually their most immediate priority is to retain possession by getting a 1st down. One way to measure a play’s success is by how much it improves or impairs a team’s probability of getting there.

The graph below illustrates the probability of eventually achieving a 1st down in a current series given various ‘to go’ distances on 2nd down and 3rd down. Data is from all regular season plays from 2002 through 2007.


We can see that the break even point for 2nd down is 5.5 yards. In other words, a team (whose purpose is to get a 1st town) should prefer a 2nd down and 5 to a 1st and 10, but it should prefer a 1st and 10 to 2nd and 6.

For 3rd down, the break even point is at 1.5 yards. A team should prefer a 1st and 10 to any other 3rd down situation 2 yards or longer. This was a little surprising to me. I expected the break even point to be around 3rd and 3 or 4.

We can also compare other situations. Consider a team with a 2nd and 8. The chance of getting a first down on their current series is 57%. To break even in terms of 1st down probability, they need to gain at least 5 yards to have a 3rd down and 3. In fact, the two probability curves are generally separated by 5 yards, between the usual situation of 10 yards to go and 2 yards to go.

Five yards appears to be the magic number. Unless a team gains 5 or more yards on a given play, it’s a setback in terms of 1st down probability.

Carroll, Parmer, and Thorn, the authors of The Hidden Game of Football, established a measure of football play success in similar terms. In their system, if an offense gains at least 4 yards on 1st down, the play is considered a success. On 2nd down, the play is considered successful if it gains half the remaining distance to the first down marker. And 3rd down is only considered a success if it gains a 1st down. This may sound familiar to some readers because the Football Outsiders blog applies this same system, with limited modifications, in their DVOA scheme.

According to the actual probabilities of gaining a first down, however, this system isn’t always consistent. An offense needs 5 yards on 1st down, not 4, to break even in terms of first down probability. Also, gaining half the yards remaining on 2nd down often leaves a team worse off than it was before the play began.

Getting anything less than a 1st down on 2nd and 4 or less leaves a team with a lower probability. On 2nd down and 7, a team’s chance of getting a 1st down is 62%. Gaining half the remaining yards leaves the offense at 3rd and 3.5 yards, which yields a 1st down 54% of the time. They’d need to be inside 3rd and 2 to break even.

This is only one way to measure success in football. There are others, such as expected points and win probability. I’ll take a look at those methods in upcoming articles.

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8 Responses to “First Down Probability”

  1. dave b says:

    Long time lurker. Cool post. I'm somewhat surprised that teams convert greater than 33% of their third downs with more than 10 yards to go and even around 20% with 15 yards to go.

    38-40% is about median team 3rd down conversion rate. I'm also curious what the yards to go distribution looks like for each down.

    Does the 66% include 1st downs that have less than 10 yards due to penalties?

  2. brian m says:

    Great stuff as usual Brian.
    Is there a significant difference in these stats as you get close to the goal line? Is it harder to make a first down from the opponents 20 than from your own 20 (for example)? It seems like there would be some effect from having less room, and maybe the defense being more intense because you are about to score.

  3. Brian Burke says:

    Yes, the 66% conversion rate includes all penalties. It's for every series of downs excluding those for which time ran out. It also includes red zone series. The rates definitely decline close to the end zone.

    But that could be for at least 2 reasons--a smaller field to defend and the attractiveness of an almost automatic 3 points (or both). Teams likely get cautious throwing the ball knowing that a turnover or sack would almost certainly take away points. The latter effect is especially strong late in close games and would tend to skew the conversion rate.

  4. Brian Burke says:

    One more clarification regarding Dave's comment--It's not that teams convert 33% of their 3rd and 10s. It's that they convert 33% of all series in which they had a 3rd and 10 at some point in the series. So in some cases teams who fail to convert on 3rd and 10 go on to convert on 4th down.

  5. Anonymous says:

    Where can one find stats like rushing or passing yards on 1st, 2nd and 3rd down?

  6. Brian Burke says:

    Unfortunately, there's no publicly available source. You'd have to compile the data yourself from individual gamebooks.

  7. Anonymous says:

    Great post. However, as far as the comment about DVOA, it seems as if this is like comparing apples to oranges. My understanding is that DVOA is a different measure of success. It compares a team's performance in a situation to the average performance across the league (i.e. what do you need to do on down x with y yards to go to perform above average). You're measure of performance (which, IMO is more interesting), is looking at things from a 1st-down perspective. So even if gaining 5 yard on 2nd and 7 doesn't increase expected 1st down rate, it is better than average and in that way it is succesful

  8. Brian Burke says:

    Thanks. According to what I can tell, DVOA is based almost completely on a system published in the 1988 book Hidden Game of Football (p. 69). The system classifies plays from scrimmage as either a "win" or "failure." The authors picked 4 yards for 1st down and 50% to go for 2nd down apparently intuitively. It appears they were pretty close, but I don't think a 4-yd gain on 1st down is can be considered a win because you've actually become slightly less likely to convert the 1st down. Perhaps the numbers looked different in '88, however.

    DVOA does this same thing, then adds modifiers called "success points" (or something similar) to account for excess yards above (or below) what would be classified as a success. Then all the success points are adjusted for opponents, then compared across teams and situations to compute the % over average. So it's the success points above or below average, not yardage itself that goes into DVOA. (At least as far as I can understand.)

    One problem I have with DVOA, and FO in general, is that these formulas and stats are "proprietary," as if they're the formula for Coke or the designs for the stealth bomber. Unless they tell people exactly what they're computing, no one can tell if it's valid or worthwhile.

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