Hi, milburn-ga:
Think of one deck as a permutation (sequential rearrangement) of the
other deck, and let's assume that by well shuffling the decks all
permutations are equally likely. The property of a permutation that
leaves no card in its original position is called being a
"derangement":
[Derangement]
http://www.delphiforfun.org/Programs/Math_Topics/derangements.htm
The fraction of permuations that are derangements depends on the
number of items being swapped. As the number of items increases, the
fraction of permutations which are derangements rapidly approachs 1/e,
or roughly 36.788%.
Mathematicians often denote the number of derangements of n items by
D_n. Recall that the number of permutations of n items (without
restrictions) is n!. It can be shown:
[Derangements and Applications]
http://www.math.uwaterloo.ca/JIS/VOL6/Hassani/hassani5.pdf
that D_n is greatest integer less than or equal to:
( n! / e ) + (1/n)
for every whole number n > 1. [For the record, D_1 = 0 of course.]
Since the fraction you are looking for is D_52 / 52!, it should be
clear that the difference between that rational number and the
transcendental number 1/e is incredibly small (on the order of
10^-70).
If some aspect of my Answer requires Clarification, please ask!
regards, mathtalk-ga |