In this situation, the numbers are pretty clear: For the Saints to have any chance of winning, they have to go for it. A decision so easy, a human could make it.
Here's the full breakdown of my calculations:
Option  Chance of converting 
Chance of winning
Before play

After play  Change  

Go for it4th and 10, the 50  28%  <1%  12%  +12%  
Field goal try67 yard kick (est.)  <1%  <1%  <1%    
Punt  <1%  <1%   
Along with some circuitry to come up with a win probability for every game situation, all you need to figure out what you should do next is an estimate of how likely you are to make a field goal or convert a first down.
My estimates for these are based on the results of thousands of similar plays, but you may think you're smarter than I am. This chart shows you how changing those estimates would change my recommendation.
Based on about 1,975 fourth downs in similar situations since 2001.
Drew Brees incomplete pass to the right intended for Brandin Cooks. Drew Brees incomplete pass to the right intended for Brandin Cooks.
To estimate a team’s chances of winning, I use a mathematical model that accounts for a whole lot of variables — including the difference in score, the time remaining in the game, and the number of timeouts each team has left. On top of that, I have models for the likelihood that a team makes a field goal and the likelihood that it will convert a first down.
By combining all of this information, I can come up with the best decision a team can make, according to math.
If you want even more details about the numbers behind my decisions, my full model is available on GitHub. Help make me better!