Thinking About Luck

From time to time, I think about the batter-pitcher match-up (or batter-defense matchup) and how best to simulate the outcomes. Some games, like Strat-o-Matic, use fixed stats on batter and pitcher cards. It doesn’t matter how good the batter is if you roll on the pitcher card, and it doesn’t matter how bad the pitcher is if you roll on the batter card. A better way would be to generate Log5 probabilities for events based on the batter and pitcher/defense. So the probabilities of the outcomes adjust for each matchup, and random number determines the outcome.

The batter-pitcher match-up is more nuanced than that, however. I like to think of it as a collision of two probability distributions. For example, a batter reaching base has to do with the ability of the batter, represented by his OBP, and the ability of the pitcher/defense, represented by the pitcher’s OBP against. One might model those probabilities with spheres. They would be painted two colors, one color representing the batter reaching base (positive outcome for the batter, negative outcome for the pitcher), and one color representing the batter not reaching base (negative outcome for the batter, positive outcome for the pitcher). So the collision of the two probability distributions could be simulated by the collision of the two spheres.

There are four possible ways the spheres could collide:

1. Batter reaches base for on both the batter and pitcher spheres.
2. Batter is out on both the batter and pitcher spheres.
3. Batter reaches on the batter sphere, does not reach on the pitcher sphere.
4. Batter is out on the batter sphere, reaches on the pitcher sphere.

Now think of the four cases like this:

1. The batter executes well, the pitcher/defense executes poorly, and the batter reaches base. Think of the pitcher throwing a hanging curve ball and the batter smashing it. In this case, the batter always reaches.
2. The pitcher executes well, but the batter executes poorly. Think of the pitcher throwing a good four-seam fastball, and the batter popping it up. Or the pitcher throws a breaking ball that looks out of the strike zone, but comes back over the plate for a K. In this case, the batter never reaches.
3. Both batter and pitcher execute well. The pitcher throws a good sinker, but the batter was ready for the pitch and golfs it.
4. Both the batter and pitcher execute poorly. The pitcher throws a fast ball down the middle of the plate, but the batter doesn’t get the barrel of the bat on the ball.

In the last two cases, sometimes the batter reaches, and sometimes he doesn’t. These are the situations where the batter hits a line drive right at a fielder, or he pops a ball into no-man’s land for a single.

Take a batter with a .380 OBP, a pitcher with a .310 OBP against, and a league average OBP of .320. Log5 tells us the batter should get on base against this pitcher at a .366 clip. So where does the .366 come from? Look at the probabilities of the four cases above:

1. Both reach: 0.080
2. Neither reach: 0.490
3. Batter and pitcher both execute: 0.300
4. Neither batter nor pitcher executes: 0.130

So .286 of the .366 OBP comes from the two situations where some luck is involved. I suspect most of it comes from the third situation, where the batter executes well. Hitters like Ichiro Suzuki, however, who beat out weakly hit balls may get more from the fourth situation.

What should be clear is that there is a lot of opportunities for luck. In 600 PA, the results of 240 of them might go either way. That represents more than enough chances for good luck to lead to a career year, or bad luck to make one wonder what’s wrong with a player.

from baseballmusings.com http://ift.tt/2ijj6zS