I’d typically assume that after most card games, the cards become at least semi-ordered in some way, necessitating shuffling. However, after a standard game of war, I can’t quite figure out how the arrangement would become less random, since the winning and losing card stay together. If they’re indeed mathematically “less random,” after the game, why?
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Probability quantifies uncertainty. When we say a system is “more random”, we mean that we have less knowledge about its state. Shuffling resets our knowledge of the system to maximum randomness, i.e. a situation in which every player has no reason to believe any permutation of the deck is any more likely than any other. (The quantity that represents the “randomness” of a distribution is called “entropy”, and a uniform distribution maximises entropy for a given set of possible states.)
Something to understand is that probability is in a sense subjective. A probability distribution represents a state of knowledge. For example, if I flip a coin and ask you what the probability is that it’s heads, you would answer 50%. But as I can see the outcome of the coin flip, from my perspective the probability can only be either 100% or 0%.
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