Is this statement true?

Question

If you shuffle a pack of cards properly, chances are that exact order has never been seen before in the whole history of the universe...

If it is, what is the explanation. I find it hard to believe.

Answer 1

Here is a site with some math to go along with it.

To answer your question, it appears to be true.

The part of the website that is of interest:

A deck of $52$ cards can be ordered in $52! = 52 \times 51 \times 50 \times ...\times 2 \times 1$ ways. This is because there are $52$ ways to choose the first card, $51$ ways to choose the 2nd, $50$ ways to choose the 3rd, etc. But $52!$ is a very large number: larger than

$8 \times 10^{67}$.

How big is this number? Well, someone shuffling a deck of cards once per second since the beginning of the universe (believed to be about $14$ billion years ago) would not have shuffled the deck more than $10^{18}$ times.

Answer 2

The probabilty that a given shuffle of $C$ cards has already appeared among $N$ (independent, uniform) shuffles is $P=1-(1-\frac{1}{C!})^N \approx \frac{N}{C!}$

Let's assume some numbers: total number of people who have lived in Earth: $10^{11}$. Assuming each person has spent 100 years of life shuffling cards three times per second (slightly generous estimate), this gives $N=10^{11} \times 100 \times 3 \times 365 \times 3600 \approx 3 \cdot 10^{19}$ total shuffles.

For $C=52$ this gives $P\approx \frac{3 \cdot 10^{19}}{8 . 10^{67}}<\frac{1}{10^{48}}$