If a coach makes decisions that hurt his team's probability of winning games, then that coach is partly responsible for his team's lack of success. This week, the media has criticized Bill Parcells' decision to go for 2 points with 12:55 minutes left in the 2nd quarter and the Cowboys up 6-5. Today in the Star Telegram, I read that Parcells will continue using the "chart" at any point in the game until someone shows him evidence that proves the "chart" shouldn't be used so early.

So, I accepted the challenge, and I found a formula that can answer the question of whether Parcells is wrong. First, we need to know the probability of winning the game when you're up by 1, 2, and 3 points with 12:55 remaining in the the second quarter. These are the possible amounts that the Cowboys would be ahead depending on the result of the conversion try. Footballcommentary.com used a backward-induction formula to find these probabilities:

Probability of winning when up by 1 point (if fail the conversion attempt): .537

Probability of winning when up by 2 points (make the extra point): .562

Probability of winning when up by 3 points (make the two-point conversion): .592

Now, these probabilties of winning must be given appropriate weight depending on the sucess rate of the extra-point and two-point conversion attempts. The following formulas assume that the success rate of an extra-point attempt is 98.5 percent and the success rate of a two-point conversion attempt is 43 percent:

1. Probability of winning if go for one point:
(.985) * (probability of winning when up by 2) + (1-.985) * (probability of winning when up by 1)

Just in case you're wondering about the math, this is just saying that 98.5 percent of the time you're going to be up by 2 because the extra point is successful 98.5 times out of 100. I add in the probability of winning for the 1.5 percent of the time when you're unsuccessful on the extra point. Plugging in the above-listed numbers for the probability of winning when up by 2 points and 1 point reveals:

(.985)*(.562) + (.015)*(.537) = .561625

That's about a 56.2 percent probability of the Cowboys winning if Bill Parcells decides to take the easy extra point.

2. Probability of winning if go for two points:

(.43) * (probability of winning when up by 3) + (.57) * (probability of winning when up by 1)

In other words, 43 percent of the time you're going to convert the two-point try and be up by 3. You also have to factor in the probability of winning if you fail (which happens 57 percent of the time). We know the probabilities for winning when up by 3 and 1, respectively, so I plug those in to find out the overall probability of winning the game when Parcells decides to go for 2 with 12:55 remaining in the 2nd quarter and his team up by 1 point:

(.43)*(.592) + (.57)*(.537) = .56065

That's about a 56.1 percent probability of the Cowboys winning when Parcells decided to go for 2.

The conclusion from all this is that Bill Parcells' decision didn't significantly affect his team's probability of winning the game on Sunday. Of course, the end result was that the Cowboys were only up by 1, which left them at a 53 percent chance of winning, as opposed to the 56 percent chance they would have had if they had simply gone for the extra point. But Parcells' decision also gave the Cowboys a chance of being up by 3 points, which would have given them a 59 percent probability of winning the game. In other words, the actual decision didn't place the Cowboys in a worse position; rather, it was the end result. One of the cardinal sins of second guessing is failing to consider the added benefits if things had gone right. You can't discount the added benefit to the Cowboys of a successful two-point conversion attempt. In case you don't believe me, this article proves that going for two points early in a game doesn't significantly affect a team's chances of winning the game: http://www.footballcommentary.com/earlygame.htm

I'd say that if you had a team that was really good at the two-point conversion attempt and could make it 60 percent of them time, it might even be a good idea to always go for 2 in the first half. Over the course of a season, you'd gain more points than by always going for the safe extra point. Where n is the number of touchdowns you score in a season: (.6)(2)(n) > (.99)(1)(n). In other words, getting those two points 60 percent of the time leads to more points than getting one point 99 percent of the time.

As a parting shot, if anyone recorded that awful game on Sunday, check out the "safety" that was called on Julius Jones in the first quarter after the Cowboys' defense made a huge 4th-and-goal stop. You'll find that Jones' knee rolled over a Redskin player' leg before finally landing outside the end zone (on about the half-yard line). Unfortunately, Big Bill didn't challenge that call, which happened to lead to the Redskins' subsequent field-goal drive. If Parcells had correctly challenged the play, he would have saved the team 5 points. To me, that looms even larger than this two-point conversion debate.

If you're a mathemetician and interested in backward induction, check out the the "Dynamic Programming Model," which provides a method for calculating a team's probability of winning at any point in a football game: http://www.footballcommentary.com/dynamicprogramming.htm. Unfortunately, there's not a convenient chart, and the model requires some complicated backward-induction calculations that are out of my area of expertise.