Re-reading my previous post on the 4th-and-2 question, I realize it may seem a bit disappointing that I failed to offer a stronger conclusion on whether or not Belichick's decision was ultimately correct. I thought it important to err on the side of understatement - to elucidate the reasons why the punt tends to be overrated in such situations without using any over-optimistic estimates that might encourage a skeptic to dismiss the thought process entirely. However, I think it is possible to make a stronger attempt at showing why 0.77 is such a realistic value for P(S), without jeopardizing the tightness of the case:
1) Heading into this week, Indianapolis has had 97 offensive possessions; according to the invaluable Drive Stats page at footballoutsiders.com, these have traveled an average distance of 39.59 yards. So P(S) is essentially the probability that they could sustain a drive covering roughly 75% of their season average. If we account for the fact that New England has allowed around 28 yards per defensive possession (e.g. by taking the average of the two figures) we still have an expected value for drive distance that is greater than that required for a TD in this case. Even though this is the mean value, not the median, this already seems like very strong evidence that any estimate of P(S) must start at values considerably higher than 50%.
2) The probability of a TD is higher than that suggested by the season yards-per-drive averages of the two teams. We must give P(S) a small boost due to the fact that Indianapolis is playing at home, on their favored artificial turf, in front of a crowd who will not disrupt their snap counts. More importantly, we must give P(S) a substantial increase because the Colts are playing with all four downs in this situation - there will be no punts or field goals. Most drives in the season-long data set would have only used three downs to try to pick up a first down, with a kick of some sort occurring on fourth down if there was no success.
3) Adjustment for the precise state of the match-up between the Colts' offense and the Patriots' defense at this point in the game should further raise P(S). Restricting ourselves to the data of the final quarter does not generate a robust sample size, of course, but it is important to take account that the match-up had clearly assumed a markedly different character late in the game. Even if we retrospectively impose justice on the proceedings by treating the bogus 39-yard PI as an incomplete pass, Manning had completed 7 of 10 fourth-quarter passes at better than 10 YPA, and three running plays had gone for 9, 11, and 4(TD) respectively. Apart from the interception thrown by Manning midway through the quarter, the closest Colts came to being in danger of failing to convert a first down was when they faced 3rd and 1 at the Patriots' 4-yard line. There was no indication that barring a bad mistake by an Indianapolis player, the Patriots could do anything to generate a significant probability of preventing a first down on a given series.
For these reasons, I fail to see how anyone can confidently maintain that P(S) was below 0.77.
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