Here are the odds of winning the division after week 3, per Football Outsiders (click to enlarge):
The Saints started 0-3 for the second consecutive year and the third time under Sean Payton. According to Football Outsiders, the Saints have about a 10% chance of winning the division. Unfortunately, that might overstate the case…per FiveThirtyEight, only about 2% of teams that start the season 0–3 make the playoffs, and 0–3 teams finish with an average of about 4.9 wins. I don’t see any reason to believe the Saints will do much better.
The Falcons, on the other hand, ascend the Mountain of Probability thanks to their in-division win on Monday. Sigh.