The Answer:

This is a fascinating question, for which there is no one answer; every game is different. About the only thing we can say for sure is that dozens of balls are used in each game.

With that said, take a look at The true life story of baseballs, an article from the Pittsburgh Post-Gazette about the number of balls used in one particular game and by the MLB in general, and reasons why balls are replaced.

—The Fact Monster