## The Importance of Goals

Dilip Vishwanat

Executive summary: The first goal is the most important.

I looked at the probability that scoring a given goal led to a win in regulation. I am using the RTSS files to pull the various leads. I went through every game from 2007-08 through 2012-13. Winning teams scored 24565 of the 36804 total goals, or 0.6674546. However, the probability that winning teams scored any particular goal goes down as the total score goes up:

 Goal Won Lost Prob 1 4658 2147 0.684 2 4532 2126 0.681 3 4133 2033 0.670 4 3639 1831 0.665 5 2768 1501 0.648 6 2054 1080 0.655 7 1273 699 0.646 8 774 408 0.655 9 393 218 0.643 10+ 341 196 0.636

So the notion that the winning team scores 66.7% of every goal goes out the window. Why?

Let's define the "value" of a goal as the amount it increases your chance of winning. Not all goals have equal value. There are 4 basic types of goals: goals that break ties, goals that extend leads, goals that create ties, and goals that cut deficits. Graphically, the value of a lead looks like:

The most valuable goals break ties or create ties. The other goals are less valuable. Goals scored in blow-outs, whether scored by the leading team or the trailing team, are essentially valueless.

Here are the various leads from these games, with the probability of winning. Leading score is the columns, trailing score is the rows:

 1 2 3 4 5 6 7 8 9 10 0 0.685 0.862 0.955 0.993 0.992 1.000 1.000 1.000 1.000 #DIV/0! 1 0.733 0.907 0.976 0.992 1.000 1.000 1.000 1.000 1.000 2 0.794 0.944 0.992 1.000 1.000 1.000 1.000 1.000 3 0.827 0.973 1.000 1.000 1.000 1.000 1.000 4 0.871 0.993 1.000 1.000 1.000 #DIV/0! 5 0.889 0.952 1.000 1.000 #DIV/0! 6 0.778 1.000 1.000 #DIV/0! 7 1.000 #DIV/0! #DIV/0! 8 1.000 #DIV/0! 9 #DIV/0!

Computing the Values

The first goal always breaks a 0-0 tie. It always creates a 1-0 lead. A 1-0 lead gives you a probability of winning of 0.684. The first goal has a value of 0.184.

After the first goal, the game has to be 1-0. The second goal can be scored by the leading team, creating a 2-0 lead, or by the trailing team, creating a 1-1 tie. A 2-0 lead increases the probability of winning from 0.684 to 0.862. It has a value of 0.178. Moving from down 0-1 to tied 1-1 increases the probability of winning from 0.316 to 0.500. It has a value of 0.184. So the calculated value of the second goal is

(0.184 + 0.178) / 2 = 0.181

The probability of winning if you score the second goal is 0.500 + 0.181 = 0.681. Exactly the observed value from the table above.

There are 4 possibilities for the third goal. The team leading 2-0 scores to make it 3-0, the team trailing 0-2 scores to make it 1-2, one team can break a 1-1 tie, or the other team can break a 1-1 tie. The calculated value of the third goal is

(0.093 + 0.129 + 0.233 + 0.233) / 4 = 0.172.

The probability of winning if you score the third goal is 0.500 + 0.172 = 0.672. The observed value is 0.170.

Powers of 2 and the Binomial Theorem

For a game of n goals, there are 2^n possible sequences of goals. For 1 goal, there are 2 sequences: 1-0 and 0-1. For 3 goals, there is one way to get to 3-0, 3 ways to get to 2-1, 3 ways to get to 1-2, and 1 way to get to 0-3. As we go up in n:

 N Scenarios Distribution of Scores 1 2^1=2 1-1 2 2^2=4 1-2-1 3 2^3=8 1-3-3-1 4 2^4=16 1-4-6-4-1 5 2^5=32 1-5-10-10-5-1 6 2^6=64 1-6-15-20-15-6-1 7 2^7=128 1-7-21-35-35-21-7-1

The right column is "Pascal's Triangle" which derives from the Binomial Theorem. The number of high value goals is the middle term(s).

 N Scenarios High Value Goals Probability 1 2 2 1.000 2 4 2 0.500 3 8 3 0.375 4 16 6 0.375 5 32 10 0.313 6 64 20 0.313 7 128 35 0.273 8 256 70 0.273 9 512 126 0.246 10 1024 252 0.246 11 2048 462 0.226

So as n goes up, the value of the nth goal goes down. The probability of a goal being "High Value" goes down faster than the value of a "High Value" goal goes up. Calculating Pascal’s Triangle says that for n=17, the probability that a goal is "High Value" is 0.185, so the value of the 17th goal is at most 0.093. For n=21, the probability that a goal is "High Value" is about 0.168, so the value of the 21st goal is at most 0.084. The limit of the value of a "High Value" goal is 0.500. At the limit, the probability approaches 0.00. 0.00 x 0.500 = 0.

As an example, suppose the score is 10-0. What is the value of scoring the 11th goal here? Zero either way. What is the probability that the winning team scores the 11th goal in this scenario? 0.500.

Retrospective versus Prospective

All of this is retrospective. Looking closely at my data, I see that the team scoring the 17th goal won 100% of the 17 goal games (1 out of 1). Does that mean that the 17th goal is actually the most valuable? Uh, no. Prospectively, we have to not only look at the value of each goal, but the probability that each goal will happen. Let's call this importance. The first goal has the most importance.

 Goal Value Probability Importance Relative 1 0.184 1.000 0.184 1.000 2 0.181 0.978 0.177 0.958 3 0.170 0.906 0.154 0.836 4 0.165 0.804 0.133 0.720 5 0.148 0.627 0.093 0.505 6 0.155 0.461 0.072 0.388 7 0.146 0.290 0.042 0.229 8 0.155 0.174 0.027 0.146 9 0.143 0.090 0.013 0.070 10+ 0.135 0.079 0.011 0.058

Value of a Goal and Average Goal Total

The value of a goal is a function of average total score. In one extreme, if the average total score was 1, the first goal would have a value of 0.500 and winning teams would score 100% of the first goal. At the opposite extreme, the total score in NBA games is up around 200 so the first point in an NBA game has a value close to zero and winning teams score the first point almost exactly 50% of the time.

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