Right, I should have said that we can never be 100% certain whether he was making a mistake or not, but given the Bayesian problem I described in my last post, it was probably much more likely that he made a mistake than that he just happened to throw 4 fastballs in a row of whatever it was. I definitely stand corrected.

If he was planning to throw the slider once every four pitches, there would still only be a 10% chance of him throwing eight straight fastballs, and I think he probably should have thrown it more frequently than that, anyway.

It’s a fiction because you can’t measure it – your measurements imply it. In physics, this equates to things like “centrifugal forces”, which only arise due to the setting of reference frames.

Jesse, my 2-1 ratio was just for illustration purposes and had nothing to do with Lidge or A-Rod. The correct ratio versus A-Rod might be 3-2 slider with no one on third base and 1-1 with a runner on third. Whatever it is, one sequence that is POSSIBLE if he is throwing optimally and adhering to the correct numbers whether they are 1-1 or 3-1 (in favor of a slider) is 4 fastballs in a row or 8 fastballs in a row. Eventually in similar situations (given enough of those situations), he will throw 8 fastballs in a row (even though that is exceedingly unlikely in any one situation) without it being a mistake. That was my point.

However, I should have said, given your point, that if we see a pitcher throw a sequence that is unlikely, given the fact that we know that pitchers DO make mistakes and do not always use optimal strategies, it is much more likely that he was not using the proper percentages and did in fact make a “mistake.”

Pascuel Perez was not the best pitcher ever, but he used to throw this weird lob of a pitch like once a game that really embarrassed hitters. I forget what the pitch was called. But it looked like a pitch any 3rd grader could hit out of the park, but since it usually came at a totally unexpected time the batter would just take it almost every time. Perez could throw a low 90s fastball for a strike, and would over and over. then he would lob that weird third grade batting practice pitch and the batter would just stand there. It always made me laugh. Problem with perez is sometimes he fell in love with making batters look stupid and he would sometimes throw that pitch a second or third time in the game. If he did then it was treated the way it deserved and boom! The pitch only worked because it was a complete surprise. As soon as one guy was embarrassed all the other players would keep an eye open for it and kill it. Moral: a third grade batting practice pitch can be very effective if thrown about 1% of the time. On the flipside, almost no pitch is effective if major league players know you will throw it every single time.

There are some cues though that maybe make the entire discussion completely irrelevant, such as the batter maybe moving up 18 inches in the box. Now if the batter is only mildly sophisticated, this will tell you something. Namely, that he is struggling to hit your splitter/sinker etc, and may well be expecting another one. This of course is an excellent time to throw a fastball.

Now if the hitter is truly sophisticated, they may step up, knowing that the pitcher might know what that means, and then sit fastball. Alternately through film study you could see other decisive tendencies in batters, and at least take a chance that the batter won’t change their approach.

It all reminds me of the Simpsons, when Bart plays Rock Paper Scissors, confidently thinking that “nothing beats rock”. Lisa of course is thinking “bart always throws rock”

In situations like that, you can throw your randomized probabilities, if you had them, out the window.