Tuesday, January 06, 2009


Silly poker odds trivia

A few interesting (though not terribly useful) observations about poker odds; I believe these are correct, though I might have missed something.

  1. A player holding pocket aces always has the best of it pre-flop when playing against three opponents, no matter what the opponents hold. Against four opponents, it's possible for the player not to have the best of it, but he can't be a very big dog (I think the worst-case is AcAd vs AhAs KcKd Tc9c 7d6d, with all the non-ace opponents holding the player's suits; that yields a pot equity of over 18.118%). If there are opponents who fold pre-flop, the cards they render dead may shift the odds. If there are six people in a hand to start, two of whom have pocket aces, and three of the players fold pre-flop, it's possible for the player who doesn't have pocket aces to be the only one to have over 33.3% pot equity.

  2. A player holding pocket aces could theoretically be almost totally dominated (less than 0.8% pot equity) when playing against nine opponents, but I believe there are only one arrangement of cards (given player's suits) which would push pot equity below 1%. If player holds AcAd, opponents hold AsAh and 6c6d-KcKd. The player has zero chance for an outright win, and less than 1.6% chance for a chop (most chops are 2-way). Changing any opponent's card adds many outs. For example, changing the 6's to 5's would allow 2345x (x > 6) as a winning out, with 126 suit combinations and 14 possibilities for x, in addition to adding some full-house outs. Without the change, the player's only chances for an out would be two-way chop with a four-of-a-kind 2-5 (remaining card anything) or else a hand with all cards in the range 2-5 (roughly 4,368 out of 201,376, though some of those double-count four-of-a-kinds or else lose outright), or a six-way chop with a straight flush (6-Q high in hearts or spades),

    It's interesting to note that even the opponent with the worst hand equity (the 7's) dominates the player by a factor of over 7.5 despite having six open overpairs; it's also interesting to note that the opponent with pocket aces has the best of it, with slightly over 10% hand equity.

  3. A player holding pocket kings against seven opponents could be drawing stone cold dead (e.g. against KK AA AA TT TT 55 55) but I don't know if there's any scenario with six opponents where the player wouldn't have any chance for even a chop. Note that the player would be drawing dead even if six of the opponents fold (any only the player with dominating aces stays in). The permutation of cards among those six players would be completely arbitrary.

  4. Against four opponents, pocket kings may have a hand equity well below 0.1%. Player holds KcKd; opponents hold AcAd, AhKh, AsKs, 8c7c. The player can win outright only with 9TJQx in clubs or diamonds (76/850,668 possible boards); the only other outs, for a five-way chop, are straight flushes (6-J high, in hearts, spades, or diamonds) or straights (8-J high, in suits that don't give opponents a flush).

Most of these observations should not be even remotely considered in actual play (it would be foolish to fold pocket aces on a ten-person table for fear that one might be totally dominated) but I find them all interesting from a mathematical perspective.

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