It's not a shot attempt. It's not looking at the box score and seeing you had 45 FGA, and assuming you had 45 possessions. Just because you have 50 possessions in a game, doesn't mean your opponent had the same either.
However, I believe this is a very important statistic to track. Just think how many games you have coached where the difference between winning and losing came down to two or three possessions. In order to give yourself every opportunity to win close games, the more possessions you can have statistically speaking, the more opportunity you will have to affect the outcome of a game.
Definition of a Possession
A field goal attempt
A missed shot, in which you do not get the offensive rebound
A turnover (should also include a team turnover)
Free-throws in a two-shot or three-shot situation
Make at least one or miss the last attempt and do not get the offensive rebound
Technical FT's and &1's do not count
Missing the front end of a 1 & 1 bonus doesn't count
Studying basketball analytics recently, I am learning that coaching staffs, mainly at the professional and college level, have Directors of Analytics, and Directors of Scouting that actually know how to write code and give real-time analytical information to bench coaches at any point during a game. Check out Syracuse University's Sport Analytics Major. Also check out Dominic Samangy, a Graduate Assistant at Arkansas. His tweets will give you a deep dive into analytics.
If you want a easy way to to figure out your team possessions, just go to Captain Calculator and plug in your numbers. However, for those math majors out there, here you go:
The Possession Formula
0.5*((Field Goal Attempts + 0.4*Free Throw Attempts – 1.07*(Offensive Rebounds/(Offensive Rebounds + Opponent Defensive Rebounds))*(Field Goal Attempts – FG) + Turnovers) + (Opponent Field Goal Attempts + 0.4*(Opponent Free Throw Attempts) – 1.07*(Opponent Offensive Rebounds)/(Opponent Offensive Rebounds + Defensive Rebounds))*(Opponent Field Goal Attempts – Opponent FG) + Opponent Turnovers))
A More Basic Formula
0.96*[(Field Goal Attempts)+(Turnovers)+0.44*(Free Throw Attempts)-(Offensive Rebounds)]
.44 multiplier has to be taken into account because not all free throws take up a possession. Technical foul shots along with “AND 1″s do not, while there might be more than two free throws on one possession. Research has determined that about 44% of all free throws take up possessions. The .96 multiplier accounts for team offensive rebounds in situations where a missed shot is tipped out of bounds by a defensive player, continuing the possession without an offensive rebound being credited. - NBA Stuffer