What are the chances of a baseball team scoring at least one run in every inning of a nine-inning game? Would it be as unlikely as a no-hitter? These questions came up at our Rotisserie League draft last week, and I volunteered that the former is much less likely. Most everyone else says I’m nuts. Here’s how I figure:
Probability of a no-hitter can be calculated by estimating the probability of NOT getting a hit and raising it to the 27th power (there would be 27 batters, none of whom would get a hit). Assuming that the “average” batting average these days is about .278 (I used the median team batting average for the American League in 2000), it follows that the probability of not getting a hit is 1-.278, or .722. Raising .722 to the 27th gives us .00015153, or once in every 6,600 games. The total number of league games played in one year is 4,860 (30 teams * 162 games), so we’d expect to see a no hitter about once every year and a half. Sounds close enough for me.
Now, let’s tackle the other part. I make the assumption that the probability of scoring in ANY inning is one in three (this is just a [ahem] ballpark figure; one in two is obviously too high, one in four seems too low). So the probability of scoring in each of nine innings is (1/3)^9 = .000050805, or once in every 19,683 games. This would be equivalent to once in every four years of league play.
I think this looks reasonable. In fact, I can’t remember watching or hearing about a game where a team scored in every inning. Maybe ESPN doesn’t know how truly unusual that would be.
So, sports fans - does this look like I’ve nailed it?
This isn’t quite right because a pitcher can face more than 27 batters and still record a no-hitter (what with walks and so forth), but he still needs to “not get a hit” from each batter he faces. For this reason, its probably not possible to calculate the probability of a no-hitter using this methodology. A pitcher could (theoreticaly) face an infinite number of batters and still record a no-hitter. Of course, the more batters he faces, the probability of a no-hitter diminishes, and as the number of batters approaches infinity, the probability approaches 0.
A more useful approach is to calculate the probabilty based on historical data for a given population; i.e., how many no-hitters have actually been recorded over time divided by the number of games. This can be used to give the expected number ofno-hitters in the futute. A similar approach could be used to see the number of games in which a team scored in all 9 innings.
I finally found a link on Google to answer your question.
On May 6th, 1999 the Colorado Rockies scored in every inning. They were the first team to do it in over 35 years and it was the ninth time it happened overall. So the OP is correct, it is much more rare than a no hitter and, in fact, more rare than a perfect game.
You don’t need to worry about walks, errors, hit batsmen, etc. because they do not count towards a batting average. Out of 27 batters who either get a hit or get out (where batting average equals the probability of getting a hit), mikeargo’s method works fine. The only issue would be if a player got on base without a hit and then got out (e.g. caught stealing), but that is rare enough to neglect. There is more error from just looking at the average team batting average. The variation in pitchers’ opponents batting average is big enough to make the probability of a no-hitter higher. Also, there is game to game variation for any given pitcher beyond random variation. Another complication would be throwing a complete game (if you mean a one-pitcher no-hitter).
Tthe basic point stands - scoring every inning is unlikely (although the White Sox got 6/9 today). Also, you wouldn’t bat in the bottom of the 9th if you were winning, and you would probably be winning if you scored in the first 8 innings. So, if you mean scoring in all 9 innings (not all 8), then it should be about half as common as your probabilities would suggest.
To figure out the probability of a perfect game, you could just look at on-base percentage instead of batting average.
No, the walks and so forth are still important because each batter faced has two possible outcomes: either a hit or not a hit. The fact that the latter outcome includes outs as well as walks is irrelevant for the purpose of calculating the no-hitter. To calculate the probabilty of a no-hitter as mikeagro has done, you take the probabilty of a single event (a pitcher facing a batter ), which would be the probability that the batter is not able to get a hit (which is different from batting average), and the multiply that probability for each such event, which would mean all batters the pitcher (or combination of pitchers) faces would count as an event, whether they are put out or not. That would give you the probability of a no-hitter.
It is also clear that according to hajario’s site, the rarity of a game in which a team scores in all nine innings is more than mikeagro’s expected outcomes would suggest; once every 19,600 games. At 2430 games a year (there are 30 teams, but they play each other, so there are only half as many games as 30*162). , that works out to once every 8 years, instead of once every 35 years. Granted, there are lot of variable factors here; the very rough guess of the probability in scoring in a particular inning that mikeagro made, the fact that the number of teams has changed over time. Also, the probability of scoring in one inning is probably not independent of scoring in another.
The point remains, scoring in all 9 innings (at least in Major League Baseball) is much more rare than a no-hitter or even a perfect game.
It’s interesting to note that no AL team has scored in every inning since the DH was instituted, something you think would make it easier to score in every inning.
Some of the problems with scoring in every inning include: uneven distribution of good offensive players throughout the batting order, increased use of relief pitchers now, and teams that are ahead by large margins not trying as hard to score.
I don’t believe that any of the teams that scored in every inning just scored one run in each inning. They’ve all put up very big numbers.
