My statistics is rusty, and even when shiny I doubt I could have done this justice. The question is inspired by a debate about hitters’ stats in baseball (specifically the value of Home Runs).
Let’s suppose you have a team where every player has exactly a .333 BA. Yes, I know how awesome that would be. However, each player hits only singles (when he gets a hit).
Haw can we estimate the average number of runs per inning/game?
I recognize there are many questions that need to be addressed, and I invite anyone interested in a puzzle like this to suggest a “rule.” For instance, 33% of the time the lead off batter will get to first. But that sets up the potential for a doubleplay. Likewise, what’s to be made of a fielder’s choice?
Let’s assume in our hypothetical there are no walks or stolen bases.
I’m just trying to get to a “well, if all you got were singles, and your team bats 333, you can expect to score X runs per game.”
If I’m asking too much, feel free to scold me. But I know I can’t do this.
I think this is FQ since it’s really stats and not sports, but if it should be in the Game Room then so be it.
I’m not sure how you would calculate such an average, because a .333 average for each player tells you nothing about whether the team’s hits are clustered together (leading to possible runs scoring) or not (unlikely to lead to runs).
Assuming nothing but singles, you would almost always need to have four hits in an inning (barring things like sacrifice flies and errors) in order to push a single run across.
I don’t think so, because it’s quite common for a man on second to score on a single.
OK, then, at least three, if not four, hits in a single inning.
OTOH, then you also have to factor in fielder’s choices (as you note), sacrifice bunts, etc.
Regardless, I think there are a lot more moving parts you would need to calculate an expected average runs/game, beyond individual batting average.
Fair. If it can’t reasonably be done, then that’s fine. But my stats isn’t strong enough to even declare it impossible. I took one semester 30 years ago. I do recall receiving an A, however
Just filing and not a problem at all, but it does fit better in Sports and tagged for Baseball.
I’m also going to page @RickJay as one of our resident Baseball stat experts. He’s really up on this kind of stuff.
I ran it as the probability of getting n number of hits in an inning with n+3 trials for total number of at bats. I made it simpler but assuming 4 hits necessary to score a run and one run per hit afterwards. Otherwise there are too many variables. (what if an atbat led to a double play? what if the runner scored from 2nd base or 1st base…, errors, walks, etc).
Take the sum of runs per inning  0.50823 times 9 innings. I did not take an 8 atbat innings into account because of playing at home and ahead.
4.574 runs scored per game.
# of hits 
Probability 
# Runs 
Probability x # of Runs 




0 
29.6% 
0 
0 
1 
39.5% 
0 
0 
2 
32.9% 
0 
0 
3 
21.9% 
0 
0 
4 
12.8% 
1 
0.128029264 
5 
6.8% 
2 
0.136564548 
6 
3.4% 
3 
0.102423411 
7 
1.6% 
4 
0.065030737 
8 
0.7% 
5 
0.037257193 
9 
0.3% 
6 
0.019870503 
10 
0.1% 
7 
0.010045643 
11 
0.1% 
8 
0.004870615 
12 
0.0% 
9 
0.002283101 
13 
0.0% 
10 
0.00104073 
14 
0.0% 
11 
0.000463372 
15 
0.0% 
12 
0.000202199 
16 
0.0% 
13 
8.67068E05 
…
There’s something odd about your table, @OldManLogan – unless I’m not understanding the “Probability” column correctly, it adds up to much more than 100%.
You’re right. I was trying to pull probability quickly using Binomial but the number of trials (atbats) changes. I accounted for the change in trials but they become independent instead of dependent events.
Let’s make a few assumptions based on the OP’s scenario;

The team’s hitters do not do any better or worse in clutch situations; they always hit .333 no matter what.

Thought they seem curiously devoid of power, the outcome of singles is the same as could be normally expected; they aren’t all infield hits or something.

The team’s runners all have average speed.

The team never draws a walk. All that can happen is hitting singles or making outs.

The team never sacrifice bunts or attempts stolen bases.

Despite their weirdness, other teams do not adjust in any way specific to Team .333.
The relationship between batter actions and runs scored is reasonably predictable and can be guessed through the Runs Created formula. As a team with a .333 on base percentage will have about 6150 plate appearances I’ll assume that, and plugging that many singles into the RC figure, I get 683 runs scored in a season, or 4.21 runs a game, which in most years wouldn’t be very good but it’s not really terrible. (IT’d be good this year; offense is way down.)
Runs created  BR Bullpen.
By the standards of 2022 this team would actually get far more runners on base than the average team  even without ever drawing a walk or being hit by pitch  but would have way less power, of course.
In 2021, the average American League team scored 745 runs; while that team got one base much less than Team .333 (the league on base percentage was .316) of course it had more power, averaging 204 homers. The White Sox had a .336 OBP, just barely above Team .333, and actually hit just slightly fewer than an average number of homers, but scored 796 runs.
OldOlds, if you take one of the formulas for Runs Created and put it in Excel, you can alter home run totals to see what a homer’s value is.
Thanks that’s excellent. I’d have guessed a higher runs/game. I’ll note that OldManLogan wasn’t too far off.
And this is why when you have a baseball stat question it is great to ask RickJay to the thread.
Pretty awesome analysis.
Well, I made some assumptions you might not have wanted inserted, such as a total lack of walks. Walks are a big deal; the average team draws perhaps three walks per game. That’d add about a hundred runs a season, more or less, making Team .333 roughly an average hitting team despite never hitting a single home run.
Neat analysis, @RickJay; thank you (and @OldManLogan) for doing some legwork.
I’m glad your actual question got answered, but for this part of your post something like WPA (With Probablity Added) may help with how valuable a certain event was in a game. Depending also on what your debate was considering “value” and “hitters’ stats”.
I’d love to see what it would look like with reasonable assumptions for things like walks (and as I mentioned, doubleplays will be increased in an allsingles situation). I was just trying to keep it simple. That’s why in the OP I said feel free to make whatever rules you like.
It started as a discussion over how important home runs are to declaring someone a good hitter vs someone who is just very consistent. “Home runs are essential to bringing the runs in” vs “slow and steady wins the race.” Like all of these things, it devolved to “Team 333* would be a shitty team”
If team .333 were pulling in 8 runs/game, I’d have been vindicated.
*We weren’t calling it team 333, but I’m totally stealing that.
I’ve always been partial to players who are really good at hitting for contact, even if they don’t have much power – guys like Ichiro and Tony Gwynn (though even Gwynn had a few seasons where he hit over 10 homers).
But, as RickJay noted the other day in the monthly MLB thread, in the current game, “small ball” (that “slow and steady” approach) isn’t how things are played now. Batting averages have steadily declined in recent years, in large part because pitching is so danged good now. Teams change pitchers much more frequently, meaning that a batter is more likely to face a pitcher who they haven’t seen yet that day, and who is throwing really hard. In addition, thanks to advanced training, most pitchers have nastier stuff than they used to.
Combine that with extensive use of the defensive shift, and it’s just harder to get singles now. Advanced stats have demonstrated to managers and GMs that what is needed now is hitting the ball in the air, and home runs, and so, many batters specifically are trying for just that. And, hence, virtually no team tries to play “small ball” now.
Well, as I noted, once you add walks your team on base percentage now vaults to something like .385, which is incredibly high; I can only find one team in modern baseball history in the 380s. Even with literally no power beyond singles, that will be a very good hitting team in most seasons, pushing 800 runs scored (about five a game.) There have been some very high scoring seasons where 800 isn’t good, but not many of them. Of course, if you did have even a little power, Team 333 would be a mighty powerhouse. Even a very modest amount of power, say 200 doubles and 100 home runs (which by today’s standards is tiny) vaults their runs scored estimate to like 9501000 runs, a staggering display of offensive might.
The basic relationship is
((Times getting on base) x (Power, in total bases)) / (Total plate appearances)
There are little adjustments made to the formula to account for steals, sacrifice flies and bunts, and little stuff like that, which have to change depending on the era of baseball you are looking at  for instance, 1910s baseball players made vastly more errors and made fewer double plays than they do now so striking out was more damaging then than it is now. But honestly just go with the above formula and you’ve got a decent estimate of what you can expect from a theoretical ballclub.
The two basic elements of scoring are getting on base and power; stolen bases help a little if you’re really good at it, but not much. That said, getting on base is more important than power, within the reasonable range of professional baseball expectations, which is why Team 333 with walks is a capable team in theory despite never hitting for power.
To put a slight bow on what was done above by @OldManLogan, the probability of getting n hits in an inning where every batter has a batting average of p is:
P(n) = \frac{1}{2} \left(n^2+3 n+2\right) (1p)^3 p^n
and the mean number of hits per innning will be:
\langle n \rangle = \frac{3 p}{1p}
(of course this is all making assumptions like no walks, no double plays, etc)