Philip Rivers’s 4-yard touchdown to Keenan Allen in the Los Angeles Chargers’ 45-10 victory last Sunday tied the N.F.L. record for consecutive completions to start a game, with 25. It was the kind of play that warranted several viewings, just to marvel at Rivers’s synchronicity with Allen.

It also harbored another distinction: the most improbable completion of Week 12.

Just how improbable was determined within seconds, by tracking devices embedded in the football, the players’ shoulder pads and the goal-line pylons, among many other places.

Those devices collect data like the quarterback’s foot speed at the time he releases the ball, how much time he has to throw and the distance to the receiver. If, for instance, a quarterback is running faster than 8 miles an hour when he throws, his chances of completing the pass drop.

The devices also collect data related to the receiver, including the space between him and the nearest defender and the distance separating him from the sideline. The closer a receiver is to the sideline, the lower the likelihood of his catching the pass.

All that data is funneled through a machine learning model, which weighs those factors (and others) against similar information about more than 35,000 pass attempts to calculate the probability of a completion. Unveiled this season on the N.F.L.’s Next Gen Stats platform, completion probability is in the nascent stages of being unpacked for greater meaning — the frequency of improbable throws toward a particular receiver, perhaps, or a measure of a quarterback’s willingness to make dicey passes.

For now, completion probability is more frivolous than illuminating, a nifty bit of trivia. But it could eventually become a scouting tool.

“On a play level, it’s kind of fun to see how hard that pass was,” said Matt Swensson, the N.F.L.’s vice president of emerging products and technology. “At another level, it tells a story of these guys and their inclination to take risks and whether or not they’re able to convert that or not.”

The quarterbacks with that inclination this season run the gamut in pedigree, from Case Keenum to Aaron Rodgers, each of whom has delivered some of the more improbable completions in the league.

But since the model values certain factors more than others, Swensson said, there is not always a correlation between a spectacular catch and a low completion probability. If a quarterback throws from a clean pocket or if the receiver catches the ball in the middle of the field, the play’s success will not be viewed as unlikely as, say, that of Keenum’s 4-yard laser on the run to Demaryius Thomas for a touchdown in Week 1.

“There’s definitely a high degree of confidence in this,” Swensson said of the assessments, “but there’s not really a magic formula.”

In the case of Rivers and Allen, their touchdown contained all the elements. Rivers faked a handoff, looked left, then rolled right, almost reaching the sideline. He was running at 14.22 miles an hour when he fired the ball 16.6 yards through the air to Allen, who caught it on the boundary of the end zone, his feet dragging inbounds as his body fell out. At the time the pass arrived, Allen was actually 0.3 yards out of bounds.

Swensson said any catch with a 20 percent probability or less would be considered “pretty difficult.” The chances of Rivers’s completing that pass were a scant 14.5 percent.

Still, it ranks as only the seventh-least probable completion this season. Here are the top five:

Week 11 vs. Ravens

1. Bengals QB Andy Dalton to WR John Ross

Ravens cornerback Marlon Humphrey covered Ross step for step down the sideline so well that he probably thought there was no chance Dalton would throw his way. Dalton did, and Ross rewarded his conviction, securing the ball against Humphrey’s arm as he came down in the end zone for a 22-yard touchdown. The primary factors contributing to the 12.3 percent completion probability, the lowest in the N.F.L. this season, were the 38.9 yards the ball traveled in the air, Ross’s 0.4 yards of separation from Humphrey and his 0.2 yards separation from the sideline.

Week 2 vs. Packers

2. Vikings QB Kirk Cousins to WR Adam Thielen

Late in the fourth quarter, with Minnesota trailing Green Bay by 29-21, Cousins dropped back to pass from his 22-yard line. Just as he got hit by defensive end Mike Daniels, Cousins released a throw off his back foot that went 42.2 yards in the air, sailing between two Packers defenders to nestle in the arms of Thielen, whose momentum carried him into the end zone as he caught it. The pass had a 12.4 percent chance of being completed, in large part because of Daniels’s proximity — 0.4 yards — to Cousins when he threw the ball and because of Thielen’s place on the field, 1.3 yards from the sideline and 1.2 yards from cornerback Jaire Alexander.

Week 2 vs. Bengals

3. Ravens QB Joe Flacco to WR John Brown

In the fourth quarter, Flacco connected with Brown on a 21-yard touchdown that defied belief. Bengals cornerback Dre Kirkpatrick was draped on Brown from the snap to the end zone, but Brown grabbed the ball over Kirkpatrick, reaching around his right arm to catch it. The completion had a 12.6 percent probability because of the 37.5 yards traveled by the ball, as well as Brown’s minimal separation from Kirkpatrick (0.7 yards) and from the sideline (0.5 yards).

Week 7 vs. Lions

4. Dolphins QB Brock Osweiler to WR Kenny Stills

Since guiding the Houston Texans to an A.F.C. South title in 2016, Osweiler hasn’t had many highlights. One, though, came against Detroit, in his second start for the Dolphins in relief of an injured Ryan Tannehill. In the second quarter, Osweiler flipped a 5-yard touchdown pass to Stills, who was running so fast toward the back corner of the end zone that he flattened a security guard facing the stands. The pass had a 12.7 percent chance of connecting because of the 31.7 yards it flew but also because of Stills’s 1.2 yards of separation from cornerback Darius Slay and how close he was to the back of the end zone when he made the catch just 0.5 yards inbounds.

Week 2 vs. Patriots

5. Jaguars QB Blake Bortles to WR Keelan Cole

This might just be the prettiest pass of Bortles’s career. Bortles, since demoted, lofted a 24-yard touchdown in the first quarter that soared over New England cornerback Eric Rowe and into the arms of Cole. The catch had a 14.1 percent possibility given Cole’s 0.8 yards of separation from Rowe and his 0.5 yards from the sideline; the 1.3 yards between Bortles and the nearest defender at the time of release; and the 43.9 yards the pass traveled in the air.