Brain Teasers
Hey Coach, That's Unfair!
Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.Probability
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 two is set to start, and the opponents are starting a right handed pitcher. The fellow asks the coach if he's 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 player's 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 player's 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.
Sure enough, game two is set to start, and the opponents are starting a right handed pitcher. The fellow asks the coach if he's 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 player's 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 player's 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.
Answer
This situation can occur due to the proportional averaging. To illustrate, the following example satisfies the player's dilemma.Rookie - LH 3/8=.375, RH 7/30=.233, Total (3+7)/(8+30)=.263
Regular - LH 58/166=.349, RH 80/356=.224, Total (58+80)/(166+356)=.264
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Comments
Rookie: 1/5(.200) against lefties; 2/5(.400) against righties.Overall.300)
Regular: 0/4(.000) against LH; 132/396(.333) against RH. Overall.330)
The key is the difference in the number of at bats for the players.
Regular: 0/4(.000) against LH; 132/396(.333) against RH. Overall.330)
The key is the difference in the number of at bats for the players.
I had a problem like this in my Statistics class a few months ago. Of course, the story was less developed, but I actually knew the answer this time.
Thanks Trill for making me feel smart. It's been awhile since I've had that feeling...
Thanks Trill for making me feel smart. It's been awhile since I've had that feeling...
I didn't think it was possible on first read - but your example is all that is needed to prove it! Very good teaser... thanks.
Great example of the classic Simpson's Paradox.
WOWZA HAA
great example of baseball stuff
good teaser
well done
well done
You should have named the player or the coach Simpson. hehe.
Example:
Rookie: LH 1/999 (about .001); RH 1/1 (1.000); Total: 2/1000 (.002)
Normal: LH 0/1 (.000); RH 998/999 (about .999); Total: 998/1000 (.99
But isn't fielding important too? Start with normal player, if hitting becomes important, switch to rookie
Rookie: LH 1/999 (about .001); RH 1/1 (1.000); Total: 2/1000 (.002)
Normal: LH 0/1 (.000); RH 998/999 (about .999); Total: 998/1000 (.99
But isn't fielding important too? Start with normal player, if hitting becomes important, switch to rookie
what the heck this shouldn't be possible
Could I suggest the writer expands the answer? This is a beautifully set up story showing how the paradox can arise in real life. But then the explanation is given in brief, purely mathematical form. If the answer went through the steps in a verbal explanation I think more readers would understand it. I know the paradox but can never remember how it works. (I'm not a statistical person)
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