Earlier this summer, before the kickoff of the Major League Baseball season, I wrote an article which sought to predict the odds that any hitters might end the shortened 2020 season with a 0.400 batting average. Through a series of Monte Carlo simulations, I found that over an 81 game half-season, the odds that any one batter might hit 0.400 was about 6% - a possibility I found very exciting!

Now, with the season well underway and nearing its ‘midpoint’ (thanks to a 60, rather than 81, game regular season), I thought it would be fun to re-investigate these chances, focusing on one particular man with the best chance of doing the dang thing: Charlie Blackmon.

## Blackmon’s Statistics Through August 23rd

So far, Blackmon has had a tremendous start to the 2020 season. He’s racked up 45 hits in 111 at-bats, good for a 0.405 batting average. Furthermore, he’s played in all 28 of the Rockies’ games thus far, and made 120 plate appearances total. Plate appearances are key here because they, not at-bats, are what qualify a hitter to become “batting title eligible”. Rule 10.23 of the official MLB rule book lays out that a hitter must record plate appearances numbering at least 3.1 times the number of games played in the season to qualify for the batting title.

Since the 2020 season is going to see only 60 regular season games, that means Blackmon needs to record 186 total plate appearances, or just 66 more than he has made thus far.

## Assessing Blackmon’s Chances of Batting .400

### Minimum Number of Plate Appearances - Smallest Sample

The easiest and most likely way that Blackmon qualifies for the batting title while simultaneously keeping his average above 0.400 is hitting the minimum number of plate appearances on the head. In this world, his existing 0.405 average accounts for nearly ^{2}⁄_{3} of his end of season average, so he can afford to dip “slightly” in the coming weeks and still pull out a 0.400 average at the end of the season.

#### Computing Hits Required

To measure the probability of that happening, I dove into the numbers:

- Currently Blackmon has 120 plate appearances and 111 at-bats
- If we apply that same ratio to the 66 plate appearances remaining, we see that Blackmon will likely have more 61 at-bats

Setting a 0.400 average as our target, we can back our way into how many hits Blackmon needs to achieve in the next 61 at-bats, which are weighted by the % of plate appearances they make up (roughly ^{2}⁄_{3} have already happened, ^{1}⁄_{3} are still to come).

```
(45/111) * (120/186) + (H/61) * (66/186) = 0.400
```

If we solve for H, we see that Blackmon needs to record 23.8 Hits in his remaining 61 at-bats, or roughly a 0.3902 batting average.

#### Deriving Probability that He Hits for that Average in Remaining Games

To compute the likelihood that Blackmon is able to hit 0.3902 or better over his remaining 61 at-bats, I can break out our Monte Carlo simulator from last post, which will use Blackmon’s lifetime career average of 0.307 to simulate ten thousand different 61 at-bat sequences. Then, with the outcomes to all of those sequences known, we can look to see how many sequences end with Blackmon achieving his “target” average of 0.3902.

After letting the simulation run, we observe Blackmon recording a batting average of 0.3902 or better in his remaining at-bats **9.13% of the time**.

### On-Pace Number of Plate Appearances

With our first option of minimum at-bats only now on the table, I think it makes sense to highlight the other side of the story. What if Blackmon keeps up his current pace of perfect game attendance for the rest of the season? With many more plate appearances likely to be recorded, it will be much harder to sustain his batting performance deeper into the season, and therefore, we should be able to measure a decreased probability of reaching 0.400 across the course of the season.

As we did before, we can compute the number of plate appearances that Blackmon will likely have remaining, corresponding number of at-bats, hits required to keep the season average over 0.400, and the likelihood of him achieving that rate:

- If Blackmon were to play 60 games with the same number of plate appearances per game as achieved thus far, he would total 257 PA, or 137 more than today
- Those 137 PAs translate to roughly 127 AB remaining
- To hit 0.400 across the entire season, with his current 0.405 in his pocket, Blackmon would need to record 50.2 hits over his remaining at-bats
- That translates to a 0.3952 average, which simulations tell us Blackmon has a
**1.36% chance**of achieving

### “Goldilocks” Number of Plate Appearances

As in all things in life, the best guess for Blackmon’s end-of-season average probably lies somewhere between the two extreme estimates presented above. I highly doubt that Blackmon will have exactly the minimum qualified number of plate appearances this year, and also think that for a player of Blackmon’s age (34), betting on perfect attendance and no injuries for the rest of the season is pretty dubious.

With this in mind, I roughly split the difference between the two plate appearance totals (61 and 127) to assume a figure of 100 remaining plate appearances, which would allow for some off-days and rest days later in the season.

- To hit 0.400 across the entire season, with his current 0.405 in his pocket, Blackmon would need to record 36.6 hits over his remaining at-bats
- That translates to a 0.3935 average, which simulations tell us Blackmon has a
**3.7% chance**of achieving

## Simulations in Summary

Minimum Qualified Case | On-Pace, Perfect Attendance Case | Goldilocks Case, Plays Most Remaining But Not All | |
---|---|---|---|

Remaining Plate Appearances | 66 | 137 | 100 |

Remaining At Bats | 61 | 127 | 93 |

Hit Required | 23.8 | 50.2 | 36.6 |

Remaining Average Required | 0.3902 | 0.3952 | 0.3935 |

% Likelihood Bat .400 | 9.13% |
1.36% |
3.70% |

## What if You Believe Blackmon’s Rest of Season Batting Will Exceed His Lifetime Average?

One of the clear assumptions in the model above is that Blackmon’s future batting outcomes will more closely resemble his demonstrated past performance (as measured by career batting average) than his recent performance. This is a tricky thing to consider though, and I think it’s fair to question whether or not this is a valid assumption and whether or not a different baseline batting average might be a better input to our simulations. Here are some of the thoughts in my head on this topic:

- I personally believe in hot streaks and cold streaks as a psychological phenomenon in baseball outside of what is measurable - sometimes when a player is “hot”, past success begets future success above and beyond what you might expect in a random walk or other stochastic processes (which you could certainly argue hitting is a form of, on the flip side, hot hand be damned)
- I also believe in the power of regression to the mean. Small sample size effects happen frequently, but the test of time almost always wears those down towards something closer to a “known range” of outcomes, unless there has been a large fundamental change. The stock market was famous described as a place where “returns can be very unstable in the short run but very stable in the long run” - I tend to think of batting averages similarly
- Changes in the hitting environment not modeled anywhere else, but likely to impact batting averages:
- The fact that baseballs around the league have been jumping off the bat for the past two seasons may suggest that Blackmon’s baseline performance expectation should also be adjusted slightly upwards
- Blackmon’s home field of Coors Field may give him a small bump due to the large proportion of games he gets to play at altitude
- More forbiddingly, Blackmon has 10 remaining games against the MLB-best Dodgers, a matchup which may stymie his attempts to rack up hits

Ultimately, I think there are convincing reasons to argue that Blackmon’s expected performance for the rest of the season could best be modeled by using a 0.307 figure (his career average), 0.360 (assuming his approach is likely to stay locked in for the next few weeks at least), or something like 0.320 (similar to high career season-best average). These differences in assumed baseline hitting ability, of course, will make a dramatic difference in the probability that Blackmon is able to maintain a 0.400 average for the rest of the season, even more so than the number of remaining at-bats.

### Grid Search and Matrix Visualization

To best represent the full range of possibilities for Blackmon’s remaining games (assuming he isn’t injured tomorrow, knocking on wood Charlie!), and allow readers to select which assumptions they feel are most appropriate, I conducted a grid search across our parameters of remaining at-bats and baseline batting average, simulating ten thousand seasons at each level to compute the likelihood that Blackmon is able to conclude the season with a 0.400 average.

Take a look and see where your judgment puts Blackmon’s odds - mine leaves me at 5.480% for now. Still a long shot, but hey, fun to have this conversation at all!