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### Batting Averages

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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
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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

.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/
```
Associated Topics:
College Probability
High School Probability

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