Sorry, but I can't help but go back to baseball stats for the next couple weeks. It is playoff time for baseball, so I can't really help it.
Clayton Kershaw had a no hitter going for a while a couple days ago. No hitters are pretty rare. I got to thinking that you could maybe estimate the chances of a no hitter. Let's say a team would normally bat 0.250 against you. That is, they would get a hit every four times at bat. What are your chances of a no-hitter? You need to retire 27 batters (3 in each of the 9 innings). The probability you retire the first batter is .75. The probability of retiring the second batter is 0.75 x 0.75. The probability of having a no hitter in just the first inning is 0.75 x 0.75 x 0.75 or 42.2%.
For the whole game, the probability of a no-hitter would be (0.75)^27 = 0.000423. Unlikely.
You can give up walks or have batters reach on errors and still have it count as a no-hitter. I don't think we need to take that into account, though, as they do not count as official at-bats anyway. I had to think about that a bit, but I'm pretty sure I'm right on that.
Then I thought about estimating how many there should be in a season or any given period of time. I went back to 1998 because that is the last year major league baseball added teams. Since then, to the present day, there have been 30 major league baseball teams. With 162 games for each team, from 1998 to 2016 (19 years) there have been 162 x 30 x 19 = 92,340 save opportunities. If we use the above probability, the number of no-hitters during that time would be: 92,340 x 0.000423 = 39.1 saves.
How many have there actually been? 49. Keep that number in mind. We'll compare other outcomes to that.
So, not bad. In fact, a lot closer than I thought it would be.
The big question mark in all this, I think, is the batting average. The overall major league average is a little higher than this, maybe 0.260. Doing the math again would give an estimate of 27.2 (lower than the aforementioned 39.1).
But maybe we shouldn't be talking about the league average. You would figure the type to get a no-hitter is a better than average pitcher. And in fact, looking at the list of those that have thrown no-hitters shows some of the best pitchers of the past 20 years - Jake Arrieta, Max Scherzer, Cole Hamels, Clayton Kershaw, and Justin Verlander. (And there have been some pitchers that had some talent, but also a good amount of luck on their side that day of their no hitter.)
So maybe the correct batting average would be 0.240 -- 55.9 no-hitters.
Or maybe a batting average would be 0.230 -- 79.6 no--hitters.
Anyway, for those somewhat interested in the topic of baseball this was an interesting math application on baseball and probabilities.
