The Math behind "The Decision"

Here is a post that uses the mathematics of football to decide whether the decision to keep the points was a good decision or not. WARNING: There be math ahead. (I'll try and keep is simple. Promise.)

Here are the assumptions we are making:

1. There was 11 minutes left on the clock when the decision was made.

2. If the penalty was accepted, the 49ers would have gotten a 1st down at the 22 yard line. For the sake of simplicity (and the ability to use Red Zone statistics), I will say they get the ball at the 20 yard line.

3. After any 49ers score, the Cowboys would have gotten the ball on the 20 yard line for the next drive.

4. The 49ers would run about 2 minutes off the clock in the Red Zone.

Here are the resources I am using for the calculations:

1. Historical Red Zone statistics (2005-2009):

Stampede Blue

These numbers are a little old, but the percentages stay relatively stable year to year. One could argue the 49ers Red Zone performance would be better under Harbaugh than previous years, but that's what the comments are for.

2. Win Probability Calculator:

Advanced NFL Stats

A simple win probability calculator that uses historical data (2000-2008) to figure out the odds of winning at any particular moment. Again, it could be argued that the 49ers and Cowboys are not average teams, but I will go with what I have.

The Math:

We are weighing two options. Keep the points or take the penalty. Each will give us a specific win probability. The job of the coach (I'd hope) is to maximize the win probability for his team in any situation.

1. Keep the points:

In this situation, the 49ers are up by 10 with 11 minutes to go and the Cowboys start the drive at their own 20 yard line. Using the win probability calculator we get:

Win Probability: 0.10
Expected Points: +0.34
First Down Prob: 0.67
TD Prob: 0.15
FG Prob: 0.08

What this tells us is that the 49ers had a 90% chance of winning the ball game at that specific moment. Any decisions the coach made (taking the penalty for example) will have to compared to this specific win percentage.

2. Take the penalty:

If coach takes the penalty, then the 49ers have a 1st and 10 on the 22 yard line with 11 minutes to go.

The easiest way to do this analysis is to use the win probability calculator and see what it gives us at this point:

Win Probability: 0.91
Expected Points: +3.94
First Down Prob: 0.67
TD Prob: 0.44
FG Prob: 0.36

Well that was easy. 0.91 > 0.90 and my analysis is over.

OK that was too easy. Let's use historical data (2005-2009) to try and gauge SF's actual Red Zone performance.

49ers Red Zone attempts: 368

49ers Red Zone TDs: 182

49ers Red Zone FGs: 127

Even in the anemic years of 49ers offense, the team scored on 84% of their red zone possessions (the rest were turnovers or missed FGs). This number seems to be in line with the rest of the league average, and would probably be higher on today's team (better coach, super accurate kicker). Using these numbers, our results after accepting the penalty is as follows:

TD: 49% of the time the 49ers would score a TD. They would have a 14 point lead and would have run 2 minutes off the clock. In this situation the Cowboys get the ball back on the 20 yard line, down 14 with 9 minutes left. The win probability of this situation is:

Win Probability: 0.03
Expected Points: +0.34
First Down Prob: 0.67
TD Prob: 0.15
FG Prob: 0.08

FG: 35% of the time the 49ers score a FG while running 2 minutes off the clock. The Cowboys get the ball back with 9 minutes and down 10 points. The win probability of this situation is:

Win Probability: 0.08
Expected Points: +0.34
First Down Prob: 0.67
TD Prob: 0.15
FG Prob: 0.08

No Score: Barring a turnover with a big return (or a pick 6), the Cowboys would get the ball back around the 20 yard line with 9 minutes and only a 7 point lead. The win probability in this situation is:

Win Probability: 0.15
Expected Points: +0.34
First Down Prob: 0.67
TD Prob: 0.15
FG Prob: 0.08

Calculating overall Win Probability:

Now we use some math to calculate the overall win probability for the 49ers (the formula is odds of the occurrence happening times (1 - Cowboys Win Probability)):

49ers Win Probability = 0.49 * (1 - 0.03) + 0.35 * (1 - 0.08) + 0.15 * (1 - 0.15) = 0.925

This number is actually higher than that of the Win Probability Calculator. I would assume that this is because we didn't factor in the pick 6 scenario but I cannot find any statistics on that right now. Then again, the odds of a pick-6 are very low. Right, Alex?


The math indicates that taking the penalty would have increased the 49ers win probability by at least 1% (possibly more). Since the job of a coach is to maximize win probability at all times, Coach Harbaugh should have taken the penalty. At least mathematically speaking.

Epilogue -- What is missing from this analysis:

This analysis doesn't take into consideration anything non-statistical. The coach's confidence in his offense/defense, how well the team is playing (or not), and so on. The comments would be a good place for such discussions.

But if you ask me, a good measure of any decision is what does the opposing team want you to do. If the roles were reversed and the Cowboys had just kicked a FG to go up 10, I would have dearly wanted them to decline the penalty since coming back on the field would have meant more time running off the clock AND good odds of them scoring at least 3 (or more likely, 7).

Whatever your opinion on this decision is, I hope Coach Harbaugh is having the same type of discussions we are having amongst his assistants and stats guys. A great coach never stops learning, especially from his mistakes and/or controversial decisions.

- Neama D.

This is a FanPost and does not necessarily reflect the views of Niners Nation's writers or editors. It does reflect the views of this particular fan though, which is as important as the views of Niners Nation's writers or editors.