The home plate umpire is responsible for making sure the ball is always as clean as possible. If it gets stained or scuffed by being fouled off in the dirt or grass or whatever, the ump replaces it. This has an unfortunate history--Ray Chapman of the Indians was struck in the head in 1920 and died shortly after. So there is no set number of pitches, but 7 would likely be the upper limit. Either the stains become too much or it gets fouled off, or knocked out of the park.
There is no definitive answer since pitches per at bat has not been a consistently kept statistic, however according to the original Baseball Abstract by Bill James, Roy Thomas fouled off 22 pitches & eventually worked the count to 3 balls. This would mean he saw at least 26 pitches (22 strikes + 3 balls + 1 ball in play)
Using an average life span of a baseball in a typical game as 6 pitches and the average number of pitches being 275 per game. The number of balls used is 46. Factoring this number out to each team in the league playing 165 games and dividing by 2 since two teams play each other each game, the number of baseballs used during a typical MLB season would about 113,850 balls per year. According to MLB, between five and six dozen are used per game (60-70 balls).
I would say about 85-90 baseballs are used in a game. I believe, if memory serves, Rawlings estimates closer to 112-120 balls per game. Average life span of a major league baseball is 7 pitches, so the actual number would depend on pitch count, foul balls, homeruns and players flipping souveniers to fans at the end of an inning.
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