The thing about being a football coach is that, when things work, you’re a genius.

When they don’t, you’re, well, something less than a genius.

The who, what and whys of each situation are not considered. Only the result is. Imagine how New Orleans Saints coach Sean Payton would be viewed if the Indianapolis Colts had recovered the surprise onside kick to open the second half of Super Bowl XLIV — which certainly would have changed the complexion of that game.

He wouldn’t be considered a genius. He likely would have been lambasted and lampooned by everyone with access to a keyboard or microphone. Maybe his career plays out differently. Maybe he wouldn’t be given the same leeway when things are rough.

Maybe everything is different. Maybe it’s all the same.

It’s impossible to know how all the hypothetical questions involved with the situation would have played out. But it’s almost certain that if the Colts had recovered the kick, many people would have sat down with a clear head and explained that odds show the kicking team has better than a 50-50 shot of recovering an onside kick — and that the recovery gave the Saints an edge in win probability.

Furthermore, if Indianapolis had recovered, it would have only damaged New Orleans’ chances to win only slightly, according to win probability statistics. But, as it turns out, the Saints recovered the ball and used the opportunity to take a 13-10 lead.

Was Payton just gambling, or does he consider win probabilities while on the sideline? It’s the latter, right?

“Absolutely,” Payton said, later adding that those decisions are about “trusting your instincts and your gut.”

While a more analytical approach has taken greater hold over the league during the past decade, it’s unlikely that mathematical equations will ever fully dictate the decisions made on the sidelines. But it’s not a bad thing that coaches are aware of the math involved, as Payton will be to some degree when the Saints visit the Chicago Bears on Monday night.

That doesn’t mean public opinion can be swayed when, after things go wrong, the value of taking a risk is supported by math.

Payton has been around long enough to know the drill. When things backfire, he’s going to take some criticism for not playing it safe. That was the case earlier this season when his team failed to convert on fourth-and-1 inside the red zone in back-to-back weeks.

After the second instance, when his team was stuffed on the goal line against the Baltimore Ravens, Payton explained the decision as being a gut feeling and said it was an important time in the game for his team. The truth is, even though he was trusting his gut, he likely was well aware of the odds, too.

According to AdvancedFootballAnalytics.com, which calculates the win probability of every play in every game, converting in that situation against the Ravens would have increased New Orleans’ odds of winning the game at that moment to 72 percent. If they failed, which they did, the odds would have remained at 50-50. If Payton had opted for a field goal, and it was successful, the Saints’ odds of winning would have climbed to 59 percent.

In other words, mathematically, the decision was sound. But as Payton noted after the contest, the bigger issue with those decisions was that New Orleans’ defense was unable to get a stop on ensuing drives and gave up points. And when your team is unable to make stops, well, it opens you up to second-guessing when gambles are taken.

Payton admitted as much after the Baltimore loss.

“Listen, that’s two times now, two weeks in a row we are not able to get it,” he said. “But then we are not able to defend it from that position on the field, so it was a gut feeling.”

Siding with the probabilities does not always pay off. Payton learned this recently, and Patriots coach Bill Belichick became the face of the win-probability discussion in 2010 when his decision to go for it against the Colts on fourth-and-2 backfired and led to New England losing the game.

The statistics show it was the right decision, and they brought fame to a 2002 research paper written by David Romer, a professor at California-Berkeley, who crunched the numbers and determined NFL coaches up to that point had made the wrong decision on fourth down an alarming amount of times.

Romer’s research almost certainly would have agreed with the decisions Payton made earlier this season. Still, even though following good math doesn’t always lead to a desirable reality, there are areas where Payton uses statistics to help form his decisions.

One of those areas is on two-point conversions.

“Typically we do not go for a two-point play until well into the fourth quarter,” he said. “How is the flow of the game going? Is it high- or low-scoring? If it’s high-scoring, it would be well into the fourth quarter. If it’s low-scoring, you might consider it earlier in the fourth quarter.”

When answering a question about whether he uses win probability to help form decisions on the sideline, Payton said he did but added there are other factors that go into pulling the trigger in critical situations.

“Fourth downs maybe is sometimes a feel, and a lot of times it is not having a play,” Payton said. “There are some times where you have got a play you cannot wait to run, so you would be more apt to be aggressive.”

In other words, there are things that can’t be measured with mathematical formulas. What happens on a football field always will be determined by the gut feelings and instincts of coaches.

But it’s also not a bad thing that odds and probabilities could be leading to having more informed guts.