By Mohin Banker
After Aaron Rodgers’ miraculous Hail Mary touchdown at the end of regulation of the NFC Divisional Playoffs against the Arizona Cardinals, the Packers had to decide whether to kick an extra point to advance into overtime or to attempt a 2-point conversion to win the game. In previous years, the obvious answer would have been to kick the (practically automatic) extra point. However, this season the NFL extended the kick distance from 17 yards to 33 yards (Source). As a result, the league’s extra point conversion percentage of 94.2% is the lowest since 1979 (Source). The increased difficulty of the kick disincentivizes attempting the extra point, prompting a stronger argument for attempting 2-point conversions. Teams attempted 94 total 2-point attempts in 2015 – significantly more than the 58 attempts from last season, even though the 2-point conversion rate changed by less than .005 (Source). The Packers in particular doubled their 2-point attempts, from 3 in 2014 to 6 in 2015.
Using statistics from the Packers’ 2015 season, we will try to determine the correct decision. Let’s assume that the probability of the receiving team winning in overtime is equal for both teams (but, Arizona’s offense against Green Bay’s defense is likely different from Green Bay’s offense against Arizona’s defense). With this assumption,
Pr(Winning in OT) = 0.5 * Pr(Receiving team winning) + 0.5 * (1- Pr(Receiving team winning)) = 0.5
Pr(E) : the probability of event E occurring
But to reach overtime, Mason Crosby had to make a 33-yard field goal to tie the game. This season, Crosby did not miss any extra points (36 attempts), or any of his 4 field goals between 30 and 39 yards (Source). When we conduct a hypothesis test on Crosby’s kicking percentage against the league percentage (94.6%, n = 1217), we find a p-value ≈ 0, indicating that Crosby’s kicking percentage is significantly different from the league average. So, opting for the extra point would give the Packers an estimated 50% probability of winning with Crosby kicking.
Over the season, the league attempted 94 2-point conversions with 45 successful attempts, a conversion rate of 0.479. The Green Bay Packers had 6 attempts and 4 completions, a conversion rate of 0.667, but from a small sample (Source). The Arizona Cardinals defense allowed 2 out of 5 conversions on 2-point attempts. Assuming that 0.667 is the true probability that the Packers convert a 2-point attempt, we can model their 2-point attempts as a binomial distribution. Calculating the standard error of the sample yields the following:
Standard error = sqrt(pq/n) = sqrt((⅔)(⅓)/6) = 0.192
We want to find the confidence level (and therefore critical value) that would produce an interval with a floor above 0.5. So:
0.667 – 0.192*Pr(t* > |t|) = 0.5
Pr(t* > |t|) = 0.870 with df = 5
t = +/- 0.172 OR 34% confidence
So, we are just 34% confident that the true conversion rate is above 0.5, or that the Green Bay Packers had an incentive to go for a 2-point conversion. With their playoff hopes on the line, the Packers made the right decision to go for the extra point, as they had a better chance of winning in OT. Unfortunately though, their odds were clearly not good enough, resulting in another Super Bowl spent on the couch for Rodgers and the Packers.