Three Card Poker Hole-Card Play: The Details

Hole-carding 3CP was one of the first advanced plays I made. When I saw a dealer who was flashing her hole-card at my local casino, I was excited and quickly computed optimal strategy and the edge. I thought this was a one-off, that I would not see hole-cards anywhere else. Imagine my surprise when I went to Las Vegas and saw dealers flashing their hole-cards everywhere. Nearly every casino used  Ace shufflers and casinos didn’t take 3CP seriously as a game protection threat.  Those days are long gone, but hole-carding opportunities for 3CP still exist everywhere.

Hole-carding 3CP has dashed many APs against its rocky shores (including me); it is tough for an AP to beat the game in practice. Errors in reading cards quickly evaporate any profit. Top APs will only play against 3CP when conditions at the game are perfect and there is no better opportunity available.

Here is the combinatorial analysis for the main game of 3CP, without seeing any hole-cards.

In particular,

• The house edge is 3.3730%
• The hit frequency (winning 1 or more units) is 44.9127%
• The standard deviation is 1.6393
• The modal (most common) outcome is to lose 1 unit.

Basic strategy for the main game is to play Q64 or higher and fold all other hands.

Here is the combinatorial analysis if the player sees one dealer hole-card before making is Play/Fold decision:

In particular,

• The player edge is 3.4829%
• The hit frequency (winning 1 or more units) is 54.2203%.
• The standard deviation is 1.7417
• The modal (most common) outcome is to lose 2 units.
• The DI (desirability index) is 19.9966.

The following table gives the edge by dealer hole-card, not taking into account the player’s starting hand. This table also presents hole-card strategy, which can be simply expressed as “Any, Q92, K92, A92.”

The most common partial-read the hole-card player gets is when he sees that the card is a paint card (Jack, Queen, King) but can’t otherwise distinguish which paint card it is. In other words, the player can tell if the card is either 2-T, JQK, or A.  Clearly if the AP sees that the card is 2-T, he should always make a Play bet. If the AP sees that the hole-card is an Ace, he should only raise with A92 or better. As the following table shows, if the AP sees that the hole-card is an unknown paint card (JQK) then he should make a Play bet with QJ5 or better:

With the QJ5 strategy, the following table gives the combinatorial analysis for hole-card play when the AP can’t distinguish paint cards:

In particular,

• The player edge is 2.4074%
• The hit frequency (winning 1 or more units) is 52.8926%
• The standard deviation is 1.7271
• The modal (most common) outcome is to lose 2 units.
• The DI (desirability index) is 13.9395

The following table gives the player’s edge, not taking into account the player’s hand, based on the observed dealer hole-card:

If an inspired reader wants to learn how to analyze hole-carding, I recommend 3CP as a good starting problem. It is just tough enough to require some careful thought. But it is easy enough that you can figure it out yourself. Moreover, you can write your program in VBA and still have it complete in a relatively brief period of time (cycle length = 1,221,511,200). 3CP was my first game. Use the numbers presented here to give it a try.