Conference rusure::math

1674.0. "Math Magazine Problem 1403" by RUSURE::EDP (Always mount a scratch monkey.) Fri Oct 09 1992 12:04

    Proposed by Richard Friedlander, University of Missouri-St. Louis, and
    Stan Wagon, Macalester College, Saint Paul, Minnesota.
    
    Simpson's aggregation paradox admits a simple baseball interpretation. 
    It is possible for there to be two batters, Veteran and Youngster, and
    two pitchers, Righty and Lefty, such that Veteran's batting average
    against Righty is better than Youngster's average against Righty, and
    Veteran's batting average against Lefty is better than Youngster's
    average against Left, but yet Youngster's combined batting average
    against the two pitchers is better than Veteran's.  Question:  Can
    there be a double Simpson's paradox?  That is, is it possible to have
    the situation just described and, _at the same time_, have it be the
    case that Righty is a better pitcher than Left against either batter,
    but Left is a better pitcher than Righty against both batters combined?

	The double paradox cannot exist.

	Suppose that the individual batting averages are as follows:

		 |LR
		-+--
		V|ab 
		Y|cd

	V's overall average can be anywhere between a & b.   Similarly, Y's 
overall average can lie anywhere between c & d.   The actual values for the
batters overall averages will depend on the relative frequency of encounters.
We are told a>c, & b>d.   Wlog, suppose that a>b.   

	For a single paradox to be possible, c must be greater than b.
Summarizing: a>c>b>d.   But for a single paradox with respect to the pitchers,
we would need a>b>c>d.   Contradiction.

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Andrew