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I'll respond with some thoughts later on, but can you explain why good hitting helps more than good pitching in the regular season?
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In the regular season, most teams employ a 5-man rotation with a 6 or 7-man bullpen. The schedule includes games against every team in the league from the World Series contending clubs down to the perennial cellar-dwellers.
In the playoffs, teams shorten their rotations to 3-4 pitchers. Their fourth and fifth starters become relievers. The two to three worst relievers get bumped off the roster entirely. This means that you'll face much better pitching in the playoffs than you do in the regular season - first, because only the elite teams will be there; second, because even within a team, only the elite starters (primarily) will face your lineup.
During the regular season, a team like the Yankees can beat up on the Oaklands and Kansas Citys by outscoring them 10-7. They can also consistently beat up on the No. 4 and 5 starters of playoff-bound teams. If you win all your games against No. 4 and 5 starters, 20% of your games against No. 1 starters, and 50% of your games against No. 2 and 3 starters, then you'll win 32 out of every 50 games (64% winning percentage), for an overall record of 103-59. Obviously, no team is going to beat the other club's No. 4 and 5 starters every single time, so let's drop 5 games from 103 for a total number of 98. Under the wild card format, 98 wins will get you into the playoffs.
Once in the post-season, you're primarily dealing with 1-3 starters. So applying the same probabilities as in the previous paragraph, you go from winning 32 out of every 50 (64%) to winning 12 out of every 30 (40%.) A 40% winning percentage isn't enough to win a playoff round, where at worst, you have to be at 60% (in a 5 game series) or 57% (in a 7 game series.) Due to sampling issues, you could also look at this as winning 11 games out of a maximum total of 19 (a more than 57% winning percentage.)
Teams built around front-end pitching improve in the playoffs because they no longer have to send out their weak 4 and 5 guys. If a team wins almost none of the games started by its 4 and 5 pitchers, and 90% of the games started by its No. 1, 2 and 3 pitchers, then there's an overall winning percentage of 54%. This is lower than the hypothetical Yankee percentage of 64%, but look at what happens in the playoffs once you exclude the No. 4 and 5 starters. The Yankees are at 40% and the team with the front-line pitching jumps up to 90%.
That's why you often see offensive-oriented clubs bounced from the first round of the playoffs. They rack up an impressive number of regular season wins but they no longer have the No. 4 and 5 pitchers to beat up on, and that dramatically changes the landscape. It's not a coincidence that the potent Texas Ranger lineups of the 90s never got out of the first round, or that recent offense-only Yankees teams have been eliminated in the first round for three consecutive years. You need at least a little bit of offense to win in the playoffs, but frontline pitching becomes a lot more important.