The “21 + 3” 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. In the original version, the payout 🏀 for each of these
was 9-to-1. With this pay table, the game has a house edge of 3.2386%. Recently, new
🏀 pay tables have been introduced that have higher house edges and greater
volatility.
The point of attack I considered is to 🏀 target flushes. Any strong imbalance
in the suits favors the player. For example, consider a situation where there are 40
🏀 cards, 10 of each suit. Without going into the math, the number of ways of making a
three-card flush is 🏀 480. Now, take those same 40 cards, and assume they are distributed
15, 10, 10, 5. Then the number of 🏀 three-card flushes is 705. The more unbalanced the
distribution of suits, the more the edge swings towards the player.
To make 🏀 use of
this, it is necessary to keep track of the number of cards in each suit that remain in
🏀 the shoe. This can be accomplished by a team of counters, each keeping track of one of
the suits (or 🏀 by a mentally gifted solo counter). The counters then compute the
difference between the most abundant and least abundant suits. 🏀 This difference is then
turned into a true count, and if that true count is sufficiently large, the player has
🏀 an edge.
I created a simulation to model using this system on a six-deck shoe game
dealt to 52 cards and 🏀 simulated one hundred million (100,000,000) shoes. This work
showed that a counter can gain an edge on approximately 3.5% of 🏀 the hands dealt (1.75
hands per shoe). The counter should make the 21+3 wager whenever the true count is 8 🏀 or
higher. The average edge when the wager is made will be just over 5%. If the table
limit isR$25, 🏀 then a counter playing heads-up can earn aboutR$2.20 per shoe. The new
pay tables were not evaluated.
As an experiment, I 🏀 shuffled one hundred thousand
(100,000) shoes and computed the edge at the point when there were 100 cards remaining
in 🏀 the shoe. The result of this simulation was an average house edge of 3.247%, which
is close to the theoretical 🏀 value of 3.239%. More interesting was that the standard
deviation of the house edge was 3.57%. It follows that a 🏀 player edge is 0.910 standard
deviations above the mean. Therefore, the player will have an edge on about 18.14% of
🏀 the shoes at that point. The trick is knowing which ones. My simulation gave a maximum
player edge of 23.71% 🏀 and a maximum house edge of 13.55%.
There are two reasons that
APs will not target 21+3 with this system. The 🏀 first is its complexity, the second is
the low return. However, there is another approach that may be significantly
stronger.Consider 🏀 a shuffle tracking approach where a slug of cards is identified that
is either deficient in one suit or abundant 🏀 in one suit. In this case, by tracking that
slug through a weak shuffle, the AP will have a good 🏀 opportunity. My knowledge of
shuffle tracking is minimal. I cannot say if this is an approach that has been used 🏀 in
practice. Finally, I have not considered if the new pay tables have a similar
vulnerability to the 9-to-1 pay 🏀 table.
For more information on this topic see:
The
following are my recommendations regarding 21+3:
