Possible and actual football scores
The home team lost in a new way yesterday. The Baltimore Ravens beat the Houston Texans by 25-9. This was the first time that score has been seen in the NFL.
Possible individual team scoresHow many scores are possible? It is possible to score any number of points except 1. You can score 2 points for a safety, so you could score any even number of points via safeties. You can score 3 points for a field goal, so you can score any odd number of points, except 1, by a field goal and as many safeties as necessary.
Actual individual team scoresThe highest score in an NFL game was 73 points. No team has scored 67, 68, 69, or 71 points. Otherwise, all possible scores up to 73 have been seen in actual games.
Possible pairs of scoresAssume a maximum possible score of M. Then there are M possible winning scores: 0, 2, 3, 4, ..., M.
There are also M possible losing scores, but there are less than M^2 possible total scores since the winning score cannot be less than the losing or tying score.
Out of the M^2 pairs of two numbers coming from a set if M numbers, M of these pairs are tied, and in half of the rest the first number is higher than the second. So the number of possible scores, with each score bounded by M, is
M + (M^2 - M)/2 = M(M + 1)/2.
If M = 73, there are 2,701 possible scores.
Actual pairs of scoresThere have been 1,076 unique scores in NFL football. (There were 1,075 until yesterday.) That means there are 1,626 possible scores we haven't seen yet (assuming the winning team scores no more than 73 points). There are 256 scores that have only been seen once.
The smallest score not yet seen is 4-0.
Here's a visualization of actual scores. The vertical axis is the winners score, from 0 down to 73, and the horizontal axis is the tie or loser score, starting from 0 on the left.
