## Tuesday, April 2, 2013

### Batting averages, plate appearances and the probability of a hit

Question:  The likelihood a batter realizes an official at bat each time he comes to the plate is 4/5.  His batting average is 0.300.  What is the probability that he gets a hit in each plate appearance?

Answer:  An appearance at the plate does not always result in an official at bat because a person can be walked, hit by a pitch, or successfully implement a sacrifice.  A hit can only occur when the batter has an official at bat.  The batting average is the conditional probability of a hit given that a plate appearance results in an official at bat.

Intuitively, the probability of a hit for all plate appearances is the weighted average of the probability of a hit given that the plate appearance ends up in an at bat and the probability of a hit given that the plate appearance does not result in an at bat.  This is Baye’s law.

P(Hit)= P(Hit/At bat)x(P(At bat) +P(Hit/No at bat) x P(No at bat)

In the current problem:

P(Hit)=(0.300)x(4/5)+0x((1/5)

=0.240

Note:  It is now baseball season so I will post several baseball math problems.

Also, please consider my book on Kindle:

The book is a bit dated and needs improvement.  It has some useful and interesting problems.  I am working on a second edition.

Author's Note:

This problem first appeared in my book Statistical Applications of Baseball, published in 1996.  It is available at a very low price on kindle.

https://www.amazon.com/Statistical-Applications-Baseball-Statistics-Sports-ebook/dp/B006M3PQWQ

Go back to baseball probability page by clicking here.

http://www.dailymathproblem.com/p/baseball.html