Part 15/16:
Through the lens of Markov chains, this problem becomes more approachable. Each shuffle changes the deck's arrangement—each a transition in the chain of possible states. Mathematical analysis shows that seven shuffles suffice to make the deck’s arrangement close enough to random that all configurations are equally likely, a result established by mathematician Bayer and Diaconis in 1992.
In real life, most people don’t shuffle seven times—often just a few. This results in decks that are only partially mixed, displaying subtle patterns and biases. Recognizing this, savvy players and dealers make sure to shuffle thoroughly, embracing the profound connection between a simple card game and deep mathematical principles.