Well done. You have ushered in the correct wording of "probability". The use of the word correlation is nonsense because there are too many variables to try to isolate whether the increase or decrease in any particular play molecule will give give success of the whole of the entire offense over the time of the game against the whole of the defense over the time of the game with all adjustments. It's not that simple. But there are a few good probabilities. Parcells used to factor these probabilities into each game. First, the amount of error that is factored into every pass attempt is greater than the amount of error involved in a run attempt. There are several extraneous and direct variables in one pass play, which increase by the distance of the pass and the amount of time for the play to complete, than in one simple running play. Parcells did look at the increased probability of keeping the ball and gaining forward progress, as well as controlling the clock, on a running play than attempting less probable plays downfield, especially slow-developing. His reasoning for running Ottis Anderson in Super Bowl XXV was that Buffalo had an explosive pass offense and he did not want to go back and forth down the field with them as each pass play delivers the ball back to the other team at a greater rate than a running play (obviously). But he noticed that Anderson was averaging 3.5 ypc in the earlier part of the game and reasoned (probability) that increasing the amount of runs increases the probability for a first down for every three downs, increases the probability for time of possession being away from the Bills, and reduces the probability for turnover error. Improving the number of running plays and effective running plays does increase the likelihood of time of possession for a team and time of possession increases the likelihood of offensive scoring (obviously). But look at the Niners in 2011. How confident were the Niners with Alex Smith (especially with the Garrett notion that longer pass plays open the running lanes underneath) with low percentage attempts? How confident were the Niners, despite a immature pass plan, with running Frank Gore play after play even though the defense knew it was likely that he would get the ball and the defense did not need to assign extra players deep to cover the Niner receivers? Did the Niners run anyway? We're there games in 2011 where the Frank Gore ran for less than 4 ypc? Less than 3 ypc? Did they still continue to run him anyway? Is there any benefit to continuing to run the ball for very few yards per carry?