Ultimate Texas Hold’em: The Known Card

Ultimate Texas Hold’Em (UTH) is a rich source of advantage play opportunities. This post considers yet another possibility — when the player knows one of the two cards he will be dealt in advance of placing his initial bet. I often refer to this knowledge as “top carding,” though the known card can be either player card.  Knowledge of one player card can be gained, for example, by edge sorting, accidental exposure, card location, skilled cutting, incidental marks on the cards and outright cheating. Whatever the means, the edge the player can get is extraordinary.

The work presented here is based on my combinatorial analysis for the main game (see this post). No specialized computer analysis was required. After all, once the initial wager has been made, the hand will be played according to basic strategy. The only advantage the AP gets is whether his known card favors the player or the house.

The following table shows the edge, based on the known card:

Note that the average edge across all known cards is -2.1850%, the house edge (last row in table). If the known card is a Ten, Jack, Queen, King or Ace, then the AP has the edge, otherwise the house has the edge. The AP exploits this knowledge by making a large wager when he has the edge, otherwise he will either make a small wager or sit out the hand.

After I completed this work, I decided to check my results against those presented by James Grosjean in Exhibit CAA (page 362). I was surprised to discover that our results don’t agree. For example, I give an edge of -56.97% if the known card is an Deuce. Grosjean gives an edge of -57.96% in this same situation. 

Here are Grosjean’s “known card” results:

Note that the average edge across all known cards is -2.359%, which is NOT the house edge (last row in table). It follows that Grosjean’s results are not correct. I do not mean this to be a “gotcha” against Grosjean — I am simply making a methodological point. These differences reinforce my conjecture that Grosjean’s results on UTH were obtained by simulation and not by exhaustive analysis. These differences are not consequential to the AP.

The following table gives the statistics for known card play against UTH:

Knowledge of one card in advance of making his initial wager is a stronger opportunity than hole-carding, even if the AP knows one dealer card and the Turn/River card. An average edge over 40% is a good day. A desirability index over 50 puts this play in the stratosphere.

The following procedures can be used to protect UTH against known-card advantage play:

• Include a “turn” before the deck is placed into the shuffler.
• Do not feed the deck into the shuffler until all players have placed their wagers for the next round. Alternatively, a cut card can be placed on top of the first packet of cards after it is ejected from the shuffler so that the top of this card is not visible.
• Watch for a player at first base who has a large spread in his bets.
• Watch for team play to sort the cards and keep them sorted.
• Watch for signaling if the shuffler is located at third base.
• Watch for dirty cards.
• Bring in fresh decks if something funny is taking place (and it’s not hole-carding).