Stolen Base

Date: 08/29/97 at 04:29:46
From: MINH VU
Subject: Modeling the stolen base

I need to create a mathematical model for a stolen base, identifying 
all variables and stating all assumptions. Initially the model should 
be as simple as possible; the complexity may be increased once a 
reasonable model has been created.

The baseball diamond is a square 90 feet on a side. A player on first 
base (the runner) can 'steal' second base if he can run to the base 
before the ball is thrown to second base. The typical sequence is that 
the player at first base walks a short distance toward second base, 
staying close enough to get back to first base if the pitcher throws 
to first; then when the pitcher (in the center of the diamond) begins 
to throw the ball to home plate the runner begins running to second 
base. The catcher at home plate then throws the ball to second base; 
the runner is 'safe' if he reaches second base before the ball, and he 
is 'out' if the ball reaches second first.

I would appreciate your help.

Date: 08/29/97 at 17:49:04
From: Doctor Barney
Subject: Re: Modeling the stolen base

Pretend that you are the runner on base. Think about all of the 
decisions you would have to make, in order. Try to identify all of the 
information that you would use to make these decisions. For each 
decision, consider all of the possible outcomes, making further 
decisions as necessary. For criteria that are beyond your own control 
(pitcher tries to throw you out, for example) assign an estimated 
probability for that criteria. For example:

  1. You are on first base. Where do you stand? This decision 
     depends on:
     a. who has the ball
     b. what other bases have runners on
     c. where the first baseman is standing
     d. where the pitcher is looking
     e. what the catcher is doing
     f. what the score is, the number of outs, the count as in 
       strikes and balls

     I've never stolen a base so I'm sure there are many I can't 
     think of.

     From this smaller model you will formulate a decision to stay on 
     base, lead off a little, or lead off a lot, probably up to some 
     easily estimated maximum.

  2. Now you must decide when to stay there, when to run quickly back 
     to first, and when to try to steal. For this decision try to 
     identify all of the factors that will influence this decision.  
     Many of the same factors we used above will apply, some will not, 
     and some new ones will.  You decide what's important
     a.
     b.
     c.

  3. In the rare event that you decide to try to steal, there are many 
     other actions you need to model. For example:
     a. Does the second baseman move to the base?
     b. Does the catcher signal to the pitcher?
     c. Does the pitcher or the catcher (or anyone else) throw to 
        second?
     d. Does the throw beat you there?
     e. Does the second baseman catch it?
     f. Does he tag you?
     g. Does the ump call it right?

     Try to think of everything you can.

Get a BIG piece of paper or use a chalk board or white board and write 
down all your ideas and how they relate to each other. Don't worry 
about putting any numbers in until after you have the overall process 
identified. Then just guess at a numerical probability for each of the 
factors you identify. Based on your personal experience, how often 
does the second baseman drop the ball?  half the time? 10 percent of 
the time? less?  Does he drop it more often when the throw is from the 
catcher than when it is from the pitcher?  Eventually you can start to 
write some equations for how these factors relate to each other. At 
this point you will need to find out from the instructor what kind of 
format he wants it in, but this should get you started.  

Have fun, and keep your eye on the ball!

-Doctor Barney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/