To shorten a fairly long and not super interesting story: it was arbitrarily chosen by a powerful team owner back around 1905 when he came up with rules for the series. He was considering five, but decided seven would be better for pitchers (since one bad game would not be as devastating). He also thought seven was a lucky number.
The league experimented with playing some nine game series in 1919-1921, but felt nine was too long, too expensive, and increased the risk of the last games being interrupted by poor late-autumn weather.
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