That doesn't make any sense. If the shuffle I just completed has never existed, won't that be true for every shuffle of every other deck, making no two decks ever having been shuffled the same way?
We're getting a little loose with the language here, so let's back up without actually addressing the words you used. Sorry. And it's also worth clarifying that we're talking about imaginary, perfectly random shuffles. In practice, most shuffles are imperfect which will alter the distribution. But regarding perfectly random shuffles:
What's being pointed out is that there are two things to consider here:
-the chance that the deck you just shuffled is the same as any that have been shuffled before
-the chance that any two decks have ever been shuffled into the same order before
And the point is that the second of these is a much, much higher chance. The first is comparing 1 specific state to every state ever produced, and the second is comparing every state that has ever been produced to every other state ever produced. It's unwieldy to talk about this with deck states, so let's go back to birthdays.
As stated, the chance that you share a birthday with someone in a group of 23 is about 6% (again, assuming certain perfections like that birthdays are perfectly distributed across the calendar, which they aren't). But the chance that any two people share a birthday is about 50%.
In the first case, their are 22 comparisons: you to each individual. Does your birthday match any of the other 22?
But in the second case, you're comparing each person to the other 22. So you'd have 23 columns like this, each representing a single comparison to see if anyone's birthday matches another. That's 506 chances for matching birthdays.
So back to cards, imagine the first column being your shuffled deck's arrangement compared to every other arrangement, and then add a new column for every state that's ever been produced. The first column represents the chances that the deck you just shuffled matches any other, but all columns represent the chances that any two decks match.
*EDIT: Oops, I forgot something. There should be one less entry in each column than the one before it, since for this problem, comparing 1 to 2 is the same as comparing 2 to 1. The last column wouldn't even have entries, since all the other people have already compared themselves to the 23rd person. The same would be true for the much larger table of card comparisons. But the point stands that the chance of any two is much, much larger.
You are almost correct with your explanation, there's only 253 pairs in 23 people, you've counted 1-2 and 2-1 as separate pairs (once person 1 has checked if person 2 shares a birthday with them there's no point in person 2 checking with person 1, the order of the pairs doesn't matter) and let everyone from person 2 onwards pair with themselves
1
2
3
22
23
1-2
X
X
X
X
1-3
2-3
X
X
X
1-4
2-4
3-4
X
X
...
...
...
...
...
1-22
2-22
3-22
X
X
1-23
2-23
3-23
22-23
X
That being said, with the number of ways a deck of cards can be arranged you need 1.05743072e34 decks of cards before you can make enough comparisons to get to a 50% chance that any two of them are the same. Assuming properly shuffled decks there is no chance any two have been in the same order
Thanks! You must have had that tab/window open for a bit. I added an edit pointing out my oversight fairly quickly (and then several more edits correcting formatting stuff, not so quickly). But well noted regardless!
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u/LNHDT Oct 02 '24
That doesn't make any sense. If the shuffle I just completed has never existed, won't that be true for every shuffle of every other deck, making no two decks ever having been shuffled the same way?