Bad Math

I heard a radio ad for a program to teach kids math in an interesting way.

They aired an example problem;

2 baseball players have the same batting average.
Alan goes 3 for 4 at his next at-bats.
Bill goes 4 for 5.
Who has the better average?

This question made me think way too hard.

The answer they want is obviously "Bill" (4/5 is greater than 3/4).
The correct answer is "not enough data".

Let's say Alan has an average of .333, previously going 10 for 30.
Bill also has an average of .333, going 20 for 60.
Alan now has an average of .382.
Bill now has an average of .369.
Alan wins.

The gap becomes pronounced when the numbers are larger.
Alan: ( 1 for 3) + (3 for 4) = .571
Bill: (100 for 300) + (4 for 5) = .341
Alan wins.

("correct" example)
Alan: ( 10 for 30) + (3 for 4) = .382
Bill: ( 10 for 30) + (4 for 5) = .400
Bill wins.

It makes me want to slap people.


A more interesting question is, starting with the same average, if they bat at different rates (3 for 4, and 4 for 5) is it possible to have the same resultant averages as each other?

I initially thought "nope, it just doesn't sound possible".

Counter-examples quickly presented themselves...
Alan: 2/ 6 = 0.333
Bill: 3/ 9 = 0.333
Alan: 5/10 = 0.500
Bill: 7/14 = 0.500

The answer is 'yes, there are an infinite number of ways to have that happen'.


Originally posted by me to LiveJournal - April 23, 2004.
Rediscovered and posted to incompetech - May 17, 2011