Buttoning up misfit pairs
I’m thinking about the Poincaré recurrence theorem, which suggests that in a finite measure-preserving dynamical system, certain states must recur infinitely often, given specific conditions. For a deck of cards, if the process is truly random and measure-preserving, considering all permutations, the probability that it eventually returns to the original arrangement is 1.
Assessing recurrence
I’m figuring out the probability of a deck of cards returning to its original order after shuffles. Since each shuffle is independent, the chance of it never returning is near zero, meaning it will almost surely return.
Comparing concepts
I’m noting that the Poincaré recurrence theorem is distinct from the properties of a Markov chain or random walk. It’s interesting to see the unique characteristics of each approach.
Assessing the scenario
Interpreting the Poincaré recurrence theorem for a deck of cards, it's almost certain to return to the original order after infinite shuffles, though the timescale for this event is extraordinarily vast.
Connecting concepts
A deck of cards, with its finite state and random shuffling, matches a measure-preserving dynamical system. This means that according to the Poincaré recurrence theorem, the deck will eventually return to the original order.
Thought about 龐加萊回歸定理與撲克牌 花的時間為 34 秒