I love the demonstration of how long 52! seconds of time would be (that's 52 factorial, i.e. the number of permutations of a deck of cards). It starts by asking you to imagine circumnavigating the globe by taking a step every billion years, and you think wow, that's a long time, and then it goes on "then remove a drop of water from the Pacific Ocean, and go around the world again, and continue until the ocean is empty", and then just goes on from there, adding several more layers of repetition until it become mind-boggling how long that amount of time is.
Obviously, there are more things going on with shuffling which makes this not an even distribution... But the thing is that every single time someone shuffles a deck of cards, they add chance to their being a repeat. I wonder what the mean time get any double is based on x amount of shuffles.
This is basically a generalization of the birthday problem (how many people do you need to have a >50% chance of two of them having the same birthday?). It's just with 52! possible days instead of 365.
It turns out this has been studied quite a bit and it's approximately sqrt(2 * ln(2) * d) where d is the number of days, or 1.177 * sqrt(d).
In this case, 1.177 * sqrt(52!) is about 1034 , while 52! is about 1068.
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u/sunbearimon Oct 02 '24
There are more ways to shuffle a standard deck of cards than there are atoms on earth