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:
Note: It is now baseball season so I will post several baseball math problems.
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The book is a bit dated and needs improvement. It has some useful and interesting problems. I am working on a second edition.