Google Answers Logo
View Question
Q: Playing cards probability. ( Answered,   0 Comments )
Subject: Playing cards probability.
Category: Science > Math
Asked by: milburn-ga
List Price: $4.00
Posted: 01 May 2004 10:55 PDT
Expires: 31 May 2004 10:55 PDT
Question ID: 339435
I have 2 decks of standard playing cards (52 cards), shuffled and
sitting side by side. I simultaneousely turn 1 card from each deck
over and see if they are identical. I repeat this 52 times. What is
the probability of, after the 52 turns, there being no match between
any of the pairs that have been turned ?
Subject: Re: Playing cards probability.
Answered By: mathtalk-ga on 01 May 2004 12:59 PDT
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


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]

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

If some aspect of my Answer requires Clarification, please ask!

regards, mathtalk-ga
There are no comments at this time.

Important Disclaimer: Answers and comments provided on Google Answers are general information, and are not intended to substitute for informed professional medical, psychiatric, psychological, tax, legal, investment, accounting, or other professional advice. Google does not endorse, and expressly disclaims liability for any product, manufacturer, distributor, service or service provider mentioned or any opinion expressed in answers or comments. Please read carefully the Google Answers Terms of Service.

If you feel that you have found inappropriate content, please let us know by emailing us at with the question ID listed above. Thank you.
Search Google Answers for
Google Answers  

Google Home - Answers FAQ - Terms of Service - Privacy Policy