In such a hypothetical situation, what the pitcher is going to throw and what the hitter thinks is coming at him are independent events.

Thus we have four scenarios (the following example is for Pitcher B - the one who throws 90% Fastball, 10% slider).

- Pitcher B is going to throw a fastball (probability = 90%, or P(PF) = .9) and Hitter C is looking for a fastball (P(HF) = .9): the probability of this scenario is P(PF) * P(HF) = .9 * .9 = .81;
- Pitcher B is going to throw a fastball (P(PF) = .9) and Hitter C is looking for a slider (P(HS) = .1): the probability of this scenario is P(PF) * P(HS) = .9 * .1 = .09;
- Pitcher B is going to throw a slider (P(PS) = .1) and Hitter C is looking for a fastball (P(HF) = .9): the probability of this scenario is P(PS) * P(HF) = .1 * .9 = .09;
- Pitcher B is going to throw a slider (P(PS) = .1) and Hitter C is looking for a slider (P(HS) = .1): the probability of this scenario is P(PS) * P(HS) = .1 * .1 = .01;

The probability of a correct guess is the sum of the probabilities for first and last scenarios, i.e. 0.82.

You can be more general and calculate the Minimum Level of Predictability given any number of pitch types a pitcher has in his toolbox, any way he mixes them, and the information the hitter has on him (that might not be accurate).

It's as simple as:

Prob pitcher throws fastball * Prob hitter looks for fastball

+ Prob pitcher throws slider * Prob hitter looks for slider

+ Prob pitcher throws curve * Prob hitter looks for curve

+ and so on...

Obviously the Minimum Level of Predictability is lower for pitchers who

- have more pitches in their repertoire
- and use them in equal proportions.

Here are some combinations coming out of my mind as examples:

# pitches in repertoire | selection percentages | MLP |

2 | 50-50 | 50% |

2 | 90-10 | 82% |

3 | 90-5-5 | 81.5% |

3 | 50-30-20 | 38% |

3 | 34-33-33 | 33% |

4 | 50-30-15-5 | 36.5% |

4 | 90-5-3-2 | 81.4% |

4 | 25-25-25-25 | 25% |

I think it's too much for theory (well, not so fast... look at the note below). Next time I'll write some real pitcher names.

Note

If you think that Pitcher P is never going to change his mixing (90% FB, 10% SL), the guessing hitter will have more success if he always guesses Fastball (he'll be correct 90% of the times instead of 82%).

Obviously, in such a case, after some time the pitcher will stop throwing fastballs to that hitter at all; then the hitter will adjust and look only for sliders. After some back and forth adjustments the couple will reach an equilibrium. Ideally that would be at 50-50, since it's the most unpredictable combo; but the pitcher, as we said in our example, might not have equal confidence in both pitches and/or one pitch might be more stressful for his body; thus he will reach a different equilibrium (90-10 in our example).

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