Blackjack is a complicated game, where subtle changes in rules or playing strategy can have asurprising impact on the player's advantage. As with many other complex real-world systems,blackjack analysis is typically done via simulation. Aside from being somewhat unsatisfying mathematically, simulation can make it difficult to explore a large number of rule and strategy variationsand rarely provides insight into the underlying reasons for the results.We have presented an analysis framework for card-counting systems that operates completelywithout simulation. This framework is based on the observation that certain aspects of the game ofblackjack have nite memory and are well modeled by discrete Markov chains. By using techniquesfrom Markov chain analysis, we developed a framework for calculating the player's long-term ex-pected gain and exercised this technique on the well-known Complete Point-Count System. Byusing our method, one could further investigate detailed aspects of the game. For example, bydetermining the reasons underlying particular advantages, a player could assess rule variations andmake strategy adjustments accordinglyAs in any case of trying to match the real world to a tidy mathematical framework, some sim-plifying assumptions were required. These assumptions introduce inaccuracies into the analysis,but we have attempted to make only the mildest assumptions required. Even then, a direct appli-cation of Markov chains produced a problem that was technically well-dened but computationallyintractable. It was only through further exploring the structure of the system that we were ableto reduce the complexity of the calculations. More accuracy could be achieved if we worked toreduce the impact of our assumptions, for example by keeping track of card counts over more thanthree categories. However, increasing the complexity in this way would quickly lead us back into aproblem that is computationally impractical.In this work, Markov chains have been a powerful tool for exploiting the inherent structurethat exists both in the game of blackjack and in card-counting strategies. Though black-jack is only a casino game, our exercise illustrates the power of Markov chains in analyzingcomplex systems. The continued development of more advanced techniques for Markov chainanalysis should only extend the utility of Markov chains for more complicated, real-world problems.