Let Me Count the Ways: High-Scoring Games May Have Unique Line Scores

In the 1915 World Series between the Boston Red Sox and the Philadelphia Phillies, the second, third, and fourth games all ended in identical scores of 2-1. Remarkably, the Red Sox won each of these games (the Phillies won the first game 3-1). Nonetheless, each of these games was unique.

The final score does not tell us in what innings the runs scored or whether the home or visiting team won. These questions are answered by examining the line score, the inning-by-inning account of the game. For example, here is the line score for the third game:

 

Philadelphia	001	000	000	— 1
Boston	000	100	001	— 1

 

Examining the line scores for this and the other 1915 games raises the question: Given a 2-1 score, how many line scores could there be? How many ways could those three runs be distributed among the nine innings? We disregard extra-inning games, because then the number (in theory) becomes unlimited.

The team scoring one run may do so in any of its nine half-innings, so we’ll say there are nine “ways” to score one run. For the team scoring two runs, there are 45 ways to do so: there are 45 distinct distributions of two runs over nine half-innings. To see this, first count the ways to score two runs assuming that the first run scores in the first inning: the second run may score in any of nine innings (the first inning on): nine ways under that condition. If the first run scores in the second inning, the second run may score in any of eight innings (the second inning on): eight more ways. And so on, to the case where the first run scores in the ninth; then the second run must also score in the ninth, and that adds one way. So the total ways to score two runs in nine innings is 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45. 

The team that scores two runs may be the visiting team or the home team, and the corresponding reverse case for the team scoring one run. The line score from the second game of the 1915 World Series will illustrate:

 

Boston	100	000	001	— 2
Philadelphia	000	010	000	— 1

 

Imagine Boston as the home team, which would “flip-flop” the line score and create a second way to have a 2-1 game with the same half-inning tallies for each team. Putting it together, the number of ways to get a 2-1 game is 9 x 45 x 2 = 810.

The fifth game of the 1915 series was another dramatic contest, ending 5-4 in Boston’s favor. There are dramatically more ways to get a 5-4 score than a 2-1 score. We can use the method above, but other statistical techniques work, too. Let’s simply summarize the ways for one team to score a given number of runs (up to nine) in a nine inning game.

 

Runs	Ways
0	1
1	9
2	45
3	165
4	495
5	1287
6	3003
7	6435
8	12870
9	24310

 

Then (with a minor exception noted) we may calculate the number of ways to get a given score: multiply the number of ways given for the run totals in question (say, under 5 and 4), and then double that product (for the home-visitor factor). In the case of the 5-4 fifth game in 1915, take the product of 1,287 and 495, then double that: you get 1,274,132. That’s right, there are well over a million ways to get a 5-4 score: over a million different line scores ending in a 5-4 game. The fifth game of the 1915 World Series was one of them.

Two points of clarification should be made. First, if the home team has the lead entering the ninth and therefore does not bat, it has the same effect on counting ways as if zero runs had scored; the calculations are not changed. Second, there does have to be a slight adjustment (which we will not do here) if the score differential is greater than four runs. Consider the example where the final score is 8-2 in favor of the home team. Then we know from the rules of baseball that the home team did not bat in the bottom of the ninth inning; the game would have ended before that. So we cannot multiply the 12,870 ways (given under 8 runs in the chart) by the 45 ways (under 2 runs) by two. That would overestimate the total ways, since some would be impossible, although that number would be small relative to the total. Note that the visiting team can win 8-2. Of course, 6-2 (for instance) is always possible (under modern rules), since the game could end on a grand slam in the bottom of the ninth.

Let’s consider the most complex case offered by our chart above: a 9-8 game. Such a game was played in the great 194 7 World Series between the Dodgers and Yankees. There are 24,310 x 12,870 x 2 ways to reach such a score: 625,739,400 ways.

This number of possibilities is almost unimaginable. Think of it this way: Suppose a team of ten (crazy) people works at writing down possible line scores for 9-8 games (without duplication) at the rate of one every thirty seconds. The team is relieved when necessary so that the task continues around the clock, ten people working constantly. lt would still take about sixty years to record all the possible 9-8 outcomes. So if you or I were not familiar with the third game of the 1947 World Series, yet tried to guess when the runs were scored, it’s unlikely we could do so in our lifetimes.

Certainly, line scores for 1-0 games have been duplicated. Most, though possibly not all, line scores for 2-1 games have occurred more than once. But what about a 5-4 contest? Or any game in which, say, nine or more runs have been scored? Have two such games ever resulted in identical line scores? The author would invite readers to report any such identical outcomes. The results presented here suggest that the vast majority of games with a considerable number of runs will have line scores unique in the history of major league baseball.