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I was recently confronted with a number—$2727707$, actually—that started a short train of thought while I was placed on hold. (This seems to happen quite often: both the observation of unusual numbers, and also being placed on hold.) Suppose you had a seven-card poker hand, consisting of all sevens, the deuces of spades and hearts, and the ten of diamonds.

Obviously, if you were playing ordinary stud, your best hand is the sevens and the ten. But suppose once all hands are dealt, the actual five-card combinations for the sake of comparison are selected randomly (with each combination having an equal probability). It is a simple, if somewhat tedious, matter to determine what the odds are that the combination in this case would be four of a kind, full house, three of a kind, two pair. (The hand cannot be a straight or a flush, and must at least be two pair.)

However, the general case is a bit more involved, in large part because the ranking of poker hands is somewhat irregular, involving a mix of sets of cards of the same rank, sets of cards of the same suit, rank sequences, and combinations of these. With that in mind:

  • Given two seven-card hands, is there any general method (that avoids brute force enumeration) for determining which one is likely to win a head-to-head matchup between randomly-selected five-card combinations?

  • In the event that there is no such general method, is there a method by which it could be quickly determined, without brute force enumeration, that a hand dominates another (i.e., that any five-card combinations of one beats any five-card combination of the other)?

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