Just a friendly reminder: David Dobbs will be speaking today at 3:35 in
HBB 102; pizza prior to that in Aconda 113.
THE PROBABILITY OF A STATISTICAL ODDITY IN BASEBALL
Professor David E. Dobbs, UT
ABSTRACT
We begin by comparing the batting performances of four baseball players.
Tables include each player’s batting averages for the first half of the
season, the second half of the season, and the entire season. It is
noticed that Player A has “lost paradoxically” to Player D in the
following sense: Player A had a higher batting average than Player D in
both halves of the season, but Player D had a higher batting average than
Player A for the entire season. The main part of the talk determines the
probability that such a “paradoxical loss” could befall Player A. The
analysis makes certain realistic assumptions. A theorem is proved
involving a branch of a rectangular hyperbola, and we also make use of the
SOLVER feature of a graphing calculator. The discussion depends only on
Precalculus and Calculus II (the latter is needed only because some
definite integrals arise in the calculations) and thus provides an
introduction to the calculation of probability for continuous (as opposed
to discrete) uniform probability density functions. We determine the
probability that Player A could lose paradoxically to a player with a
lower firsthalf batting average, as well as the probability that Player A
could lose (not necessarily paradoxically) to such a player. We close with
some comments about more realistic models, some philosophic musing about
paradoxes, and a theorem that reinforces one’s intuition about fractions.
Conrad Plaut
Professor, Director of UT Math Honors
Math Department
Aconda Ct. 104
1534 Cumberland Ave.
University of Tennessee
Knoxville, TN 379960612
Office: Aconda Ct. 401A
Phone: 8659744319
http://web.utk.edu/~cplaut
