A higher average in every split, lower average overall

Question

Two baseball hitters are compared over a season, split by the handedness of the pitcher.

Derek: 8 hits in 10 at-bats against lefties (.800), and 30 hits in 100 at-bats against righties (.300).
Evan: 75 hits in 100 at-bats against lefties (.750), and 2 hits in 10 at-bats against righties (.200).

Derek out-hits Evan against lefties and against righties. Yet Evan has the higher overall batting average.

Show that Evan's combined average beats Derek's, and give Evan's combined average.

Accepted Answer

Within each split, Derek wins: against lefties .800 to .750, and against righties .300 to .200. Now combine each hitter's own at-bats. 	ext{Derek} = \frac{8+30}{10+100} = \frac{38}{110} \approx .345. 	ext{Evan} = \frac{75+2}{100+10} = \frac{77}{110} = .700. Evan wins overall, .700 to .345, despite losing both splits. The cause is the workload mix. Derek faced mostly righties (100 of his 110 at-bats), where averages are lower for both players, so his season number is pulled down toward his weaker split. Evan faced mostly lefties (100 of his 110), where hitting is easier here, so his season n…

A higher average in every split, lower average overall

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