21 3 Blackjack odds

The 21 + 3 (TP3) blackjack side bet is based on examining the player’s two cards and the dealer’s up-card. If the three cards form a flush, straight, three-of-a-kind or straight flush, the player wins. The payout for a winning TP3 wager is a flat 9-to-1 and the game has a modest house edge of 3.239%. Recently, several casino sources expressed concern about advantage play against TP3. I then saw the following post on a web site frequented by APs, Norm Wattenberger’s Blackjacktheforum.com

This was enough to inspire me to go back and have a second look. I decided to run a “perfect play” analysis of TP3 to see if these concerns had merit. Perfect play analysis involves the following steps for a six-deck shoe:

• Go through a shoe one hand at a time.
• Before each hand, compute the exact edge for TP3 based on the cards remaining in the shoe.
• Only make the TP3 wager when the player has the edge, and at no other time.
• Record what happened.

By simulating a bunch of shoes, I was able to get results that give an absolute upper bound for beating TP3 using a method where the player does not have any prior information about the shoe (e.g shuffle tracking). I ran 10, 000 shoes and got the following results for perfect play, with the cut card placed at 260 cards (one deck from the end of the shoe):

It is interesting to break this down by the particular location in the shoe a wager is made. The following table gives perfect play statistics by the position in the shoe at which the first card of the hand is dealt:

For example, half-way through the shoe (156 cards), the AP will have the edge on 5.241% of the hands at that point, and his average edge when he makes a TP3 wager is 1.292%. If the shoe is dealt deeply, then with 39 cards remaining, the AP will have the edge 31.899% of the time, and his average edge is 9.997%.

What is important to remember is that these are “perfect play” results. The only way to achieve these numbers in a casino would be to use a computer application, live and in real-time, at the table. It is unreasonable to expect that a player would consider cheating (using a device) for a profit of 0.2748 units per shoe, especially given the low denomination of wagers usually played, and the practical difficulties involved in developing and using such a device. It is, nevertheless, important to consider this possibility in light of the recent attack on the baccarat Lucky Pairs using a device, as discussed in this post.

It is worthwhile to consider how well a human-feasible card counting method fares compared to perfect play. I previously wrote about card counting TP3 in this post. In that post, I described an approach to counting TP3 using the “suit count.” A team of counters keeps track of the suits. The counters then consider the difference between the most abundant and least abundant suits. This difference is then converted to a true count, and if that count is sufficiently large, the AP has the advantage.

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