Will a baseball player's batting average always change when the player is at bat? That is, under what conditions will the batting average stay the same? Let \(a\) be the number of times at bat, and let \(h\) be the number of hits. Then the batting average is \(\frac{h}{a}\). If the player gets a hit, then the new batting average is

\[\frac{h+1}{a+1}\]

and if the player does not get a hit, then the new batting average is

\[\frac{h}{a+1}\]

Data from Research Problem 1

A player's batting average really isn't simply the ratio \(\frac{h}{a}\). It is the value of \(\frac{h}{a}\) rounded to the nearest thousandth. It is possible that a batting average could be raised or lowered, but the reported batting average might remain the same when rounded. Write a paper on this topic.