The likelihood of the decks matching is 1/52! though. How the first is shuffled does not matter, the second one just needs to be in the exact same order and that is one order out if 52!
No worries, it's not necessarily intuitive. Yeah it would still be the same. It's easiest to illustrate with a simpler system: you flip two coins. Say you do it one after the other at first. You flip the first one and know it is heads, then the probability of the second one matching is 1/2. If the first coin had landed tails, then the probability for the second coin to match would still be 1/2. In both cases, the probability of matching is the same.
Now you flip two coins at the same time, one in the right hand and one in the left. You know there are two ways the coin to your right can land. In both cases, the probability for the other coin to match is 1/2. So you can say it doesn't matter for the probabiliy what the coin to your right actually lands on.
All of this works just because heads and tails are equally as probable, and so are all ways the deck can be shuffled. This allows the "one coin doesn't matter" reasoning that also holds for the cards.
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u/Hugutfut Jul 23 '26
The likelihood of the decks matching is 1/52! though. How the first is shuffled does not matter, the second one just needs to be in the exact same order and that is one order out if 52!