When the Patriots opted to attempt the 47 yard field goal on 4th and 18 at the end of the game, they were making the statistically incorrect call.
Over the past five seasons, the average team has converted on 12.3% of 3rd or 4th down plays where the team needs 18 yards to go; one would expect the Patriots to be slightly above average in this situation, but going for the first was a low chance of success.
Once converting a first, the Patriots would have the ball on near the 10 yard line. The Packers allow a touchdown on 60.7% of drives that make it this far, while the Patriots score a touchdown on 77.1% of these plays. For a quick an dirty calculation, averaging the two yields a rough 68.9% chance of a Patriots touchdown.
This give the Patriots 8.5% odds of a touchdown if they go for it.
Playing for the field goal, hoping to force a third down stop, and then trying for another field goal? The odds of success are even lower.
Prior to the miss, Patriots kicker Stephen Gostkowski was converting 47 yard attempts at roughly an 85.7% rate. The Packers were forced into a three and out on just 13.6% of their drives this season, with the Patriots unable to force a single three and out on the day. The odds of completing both the field goal and the stop is a mere 11.6%.
Let's say the Patriots get their three and out on the Packers after the Patriots convert the field goal attempt. The subsequent Packers punt would likely give the Patriots the ball back around their own 20 yard line, using Packers punter Tim Masthay's average punts as a reference. The Patriots are averaging just 50% of their drives that start around the 20 that end inside the opponent's 30 yard line. The odds of completing the sequence and getting back to take another 47 yard attempt is a mere 5.1%.
Going for an onside kick instead of forcing a Packers 3 and out is less likely (onside recovery is 10.2%, versus the 13.6% rate of 3 and outs).
The is no breakeven field goal distance, where the final field goal distance can justify the odds. It's just a far less likely scenario to try and win the game.
The way the Packers were moving the ball, it would have made sense to put more trust in the offense than the defense. This is a play sequence the coaches might want to have back.