Batting Averages

Date: 05/08/2001 at 23:32:30
From: Robb
Subject: Law of Large Numbers and The Gambler's Fallacy

I have a question concerning the Law of Large Numbers and the 
Gambler's Fallacy as related to baseball player batting averages.

A player has a lifetime batting average of .300 but is currently 
batting .100 after 100 attempts. Would we expect that player to hit at 
a rate greater than .300 for the remaining 500 attempts in order to 
approach his lifetime average, OR would we expect that player to hit 
at his lifetime average for the remaining 500 attempts, bringing him 
in at less than his lifetime average? Why?

Thanks,
Robb

Date: 05/09/2001 at 14:03:20
From: Doctor TWE
Subject: Re: Law of Large Numbers and The Gambler's Fallacy

Hi Robb - thanks for writing to Dr. Math.

The Gambler's Fallacy states that if each event is independent, then 
the probability of future events is unaffected by past events or 
trends. (Gamblers frequently assume that if an event has not occurred 
in a long time, it is "due," and has a higher probability of occurring 
on the next attempt.) For example, if a fair coin is tossed repeatedly 
and comes up heads five times in a row, the probability is still .5 
(50%) that tails will come up on the next toss. A gambler might assume 
that since in six tosses of the coin, it should come up tails three 
times on average (and it has not come up even once yet), it must be 
more likely to come up tails on the next toss. In other words, the 
gambler falsely assumes that the event probability changes to reach 
the long-term average instead of the other way around.

There are many reasons why the Gambler's Fallacy may not apply to this 
situation. If the player is very young, with limited experience at 
this level of play, we might expect him to hit better than his 
lifetime average as he gets better with experience. On the other hand, 
if the player is old, we may expect his skills to decline with age - 
ball players can't play forever. If the rules or conditions of the 
game have changed (such as the change in the called strike zone in the 
Major Leagues this year), we may expect that to have an impact - 
either positive or negative - on the player's batting average. And so 
on.

But all that aside, if for simplicity we assume that there are no 
effects of aging, experience, changing conditions, etc., the Gambler's 
Fallacy would predict that he will bat (approximately) .300 for the 
remaining 500 at-bats, and thus his average for the year would be less 
than his lifetime average. Specifically, his average for the year 
should be about:

     .100*100 + .300*500     10 + 150
     -------------------  =  --------  =  .267
          100 + 500            600

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/