Take a standard deck of playing cards. Give it a good set of shuffles (six or seven should do). Congratulations! The order of your 52 cards has never been created before in human history.
Of course, I can’t guarantee that your deck’s order is unique, but I can be pretty darn sure. The number of unique orderings of a deck of 52 cards is expressed mathematically as 52! (52 factorial). That’s 52 times 51 times 50 times 49 … all the way down to 1. This turns out to be a remarkably large number: 8.07 x 1067. There aren’t many numbers in the universe bigger than that (for example, it’s way more than the number of atoms that make up the Earth).
The 52-card deck has been around for about six or seven hundred years. Let’s be generous and assume that every human being alive over the past 700 years shuffled a deck 10 times a day (I think that once a day might be more accurate, but like I said, I’m being generous). So we have:
- Average number of people alive at any moment in the past 700 years: 1.2 billion
- Number of days in 700 years: 255,675
- Shuffles per day, per person: 10
1,200,000,000 x 255,675 x 10 = 3,068,100,000,000,000. Three quadrillion.
Which is a lot of shuffles. 3 x 1015. But it’s nowhere near 8.07 x 1067. To calculate the odds that your next shuffle will match any of the previous 3 quadrillion shuffles, just divide the two numbers. The result is one in 2.69 x 1052. Or
1 in 26,900,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
So yeah, it’s pretty safe to say that your shuffle is unique.
That’s just the odds of the “next” shuffle matching any of the previous ones. What about the probability any one of those 3 quadrillion shuffles matches some other one? If we compare every single shuffle with every other one, throughout history, has there ever been two that match?
Nope. The math is somewhat complicated, but the odds turn out to be, effectively, zero. Throughout history, every time a deck of cards is shuffled the resulting order has never been seen before.
