7

We have a deck with $n$ cards enumerated $1,2,\ldots,n$. The deck is shuffled. What is the probability of exactly one card to remain on its original position? What is the limit as $n$ rises to infinity?

$$ \begin{array}{rcl} \{1\} & : & \dfrac 11 \\[6pt] \{12,21\} & : & \dfrac 02 \\[6pt] \{123,132,213,231,312,321\} & : & \dfrac 36 \\[6pt] & \vdots \end{array} $$

At $n = 100 0$ and $10 000$ trials:

$$ \begin{array}{rcl} \text{value of }n & & \text{probability} \\ \hline 1000 & & 0.3739 \\ 1001 & & 0.3689 \\ 1002 & & 0.3722 \\ 1003 & & 0.3638 \\ 1004 & & 0.3707 \\ 1005 & & 0.3664 \\ 1006 & & 0.3616 \\ 1007 & & 0.3728 \\ 1008 & & 0.3702 \\ 1009 & & 0.3801 \end{array} $$

At $n = 100 000$ and $10 000$ trials:

$\text{value of } n \quad \text{probability}$

$\quad 100000 \quad \quad 0.3659$

$\quad 100001 \quad \quad 0.3552$

$\quad 100002 \quad \quad 0.356$

$\quad 100003 \quad \quad 0.367$

$\quad 100004 \quad \quad 0.3738$

$\quad 100005 \quad \quad 0.3647$

$\quad 100006 \quad \quad 0.3654$

$\quad 100007 \quad \quad 0.3637$

$\quad 100008 \quad \quad 0.3718$

$\quad 100009 \quad \quad 0.3708$

Apparently, probability approaches $0.36-0.38$, but how can one derive it analytically?

Daniel R. Collins
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Stepan
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2 Answers2

15

To find the number of "good" permutations fix one card and derange the rest $(n-1)$ cards. This can be done in $!n$ ways. ($!n$ is the number of derangements of $n$ objects). Then the number of "good" permutations is $n\cdot !(n-1)$. Hence we have:

$$\lim_{n \to \infty} p_n = \lim_{n \to \infty} \frac{n\cdot !(n-1)}{n!} = \lim_{n \to \infty} \frac{!(n-1)}{(n-1)!} = e^{-1}$$

The last limit can be seen in the link I added above.

Stefan4024
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  • Isn't this overcounting the cases where two or more cards are in their original position? – Taemyr Jan 25 '17 at 09:32
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    @Taemyr The term "derangement" means that the other n-1 cards have a position different from their original one. This is why it's $!(n-1)$, not $(n-1)!$. – Antitheos Jan 25 '17 at 11:08
11

By the inclusion-exclusion principle, in the symmetric group $S_n$ there are $$ n!\sum_{k=0}^{n}\frac{(-1)^k}{k!} $$ permutations without fixed points (see derangements). It follows that in the same group there are $$ n\cdot (n-1)!\sum_{k=0}^{n-1}\frac{(-1)^k}{k!} $$ elements with exactly one fixed point, and the limit probability is $\color{red}{\large\frac{1}{e}}$ in both cases.

Jack D'Aurizio
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  • Then the limit probability of at most one fixed point is $\frac{1}{e}+\frac{1}{e}=\frac{2}{e}$. And for two fixed points? – Jeppe Stig Nielsen Jan 24 '17 at 22:32
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    @JeppeStigNielsen: correct. The probability of having exactly two fixed points is given by $$\frac{1}{n!}\binom{n}{2}(n-2)!\sum_{k=0}^{n-2}\frac{(-1)^k}{k!}\to\color{red}{\frac{1}{2e}}$$ and the limit probability of having exactly $k$ fixed points is $\frac{1}{k! e}$. – Jack D'Aurizio Jan 24 '17 at 22:35
  • Yes, I was just realizing that. Thanks. – Jeppe Stig Nielsen Jan 24 '17 at 22:41
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    In other words, as $n \to \infty$ the number of fixed points of a random permutation of $n$ elements converges in distribution to a Poisson(1) random variable. – Michael Lugo Jan 25 '17 at 00:08