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Hey coach, that's unfair! (Posted on 2003-01-05) Difficulty: 3 of 5
A first baseman is fortunate enough for his team to be playing in the World Series. When game one is about to start, he asks the coach if he's going to play. The coach responds "despite the fact that you have a higher batting average than our rookie first baseman, we're up against a left handed pitcher today, and he has a better average against lefties than you, so I'm going to play him". Well, the fellow figures that this is fair enough, baseball being a game of averages and all, and happily sits out the first game, knowing that the team will come up against a right hander at some point, giving him a chance to play.

Sure enough, game 2 is set to start, and the opponents are starting a right handed pitcher. The fellow asks the coach if hes going to play today. The coach responds "well, I know that you have a better average overall, but today we're facing a rightie, and our rookie has a better average against righties than you do, so we're going to play him today".

So, the regular player, who has a better average against pitchers in general, has a lower average against BOTH left and right handers????? The player feels cheated. How did this happen?

For reference, the players batting average is calculated using the following formula:
average = safe hits/at bats, and is recorded to three decimal places (though announcers generally multiply this fraction by 1000 to give a integer value). A good players average will be between .300 and .350, with higher averages possible, but rare. For example a player gets 20 safe hits in 80 "at bats" then his average is .250

See The Solution Submitted by Cory Taylor    
Rating: 3.1818 (11 votes)

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Solution A simple example with explanation (solution) | Comment 7 of 8 |
It is simply due to a different proportion of attempts for the left and right hand averages. The rookie has higher averages for left and right independently, but the veteran's average for one hand is greater than the rookie's average for the OTHER hand. And the veteran has a greater proportion of attempts for that hand, thus making the overall average lean towards that average.

Here is an example:

   Left handed - 2/10 = 0.200
   Right handed - 5/10 = 0.500
   Overal - 7/20 = 0.350

   Left handed - 1/10 = 0.100
   Right - 400/1000 = 0.400
   Overall - 401/1010 = 0.397

So the rookie has higher averages for the left and right hand, individually. But the veteran's overall average leans heavily toward the right hand average because he has so many more attempts there. Because of this difference in proportion of left hand to right hand attempts, the overall averages are weighed differently, as shown above.

  Posted by Viet on 2005-06-17 01:31:31
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