Easy (yet Distressing) Answers...
Overkill,
For the 21+3 sidebet, the rank of the best first card is a ten-way tie among 3, 4, 5, 6, 7, 8, 9, 10, J, and Q. Second-best is a three-way tie among K, A, and 2.
The answer above can be simply explained. First of all, without card counting, any initial card has exactly the same probability of forming a 3 of a Kind, because the initial 6D shoe has 24 of each rank. Thus, for a 6D shoe, the 3oaK probability is just (312/312)*(23/311)*(22/310) = 0.0052484...
Similarly, any initial card has exactly the same probability of forming a flush, since the 6D shoe contains exactly 78 cards of each suit. Thus, for a 6D shoe the Flush probability is just (312/312)*(77/311)*(76/310) = 0.0606990...
Therefore, the only payout category where the initial rank matters is the Straight, because SOME initial cards provide the possibility of forming three straights, while others can be part of only two straights. In particular, any rank 3-Q gives three straight possibilities: the initial card can end up as the lowest card in the straight, the middle card, or the high card. However, for a K, A, or 2, the initial card can only form two straights: for instance, if the first card is a K, the only straight possibilities are Q-K-A and J-Q-K: the K cannot be the "low" card in the straight.
This explanation shows why for the 21+3 sidebet, the rank of the best first card is a ten-way tie among 3 through Q, and the second-best is a three-way tie among K, A, and 2.
No rank as a "guaranteed first card" provides a +EV 21+3 sidebet wager.
On the other hand, if you knew that you would get an Ace or an X as your first card, you could simply eschew the SB and bet the farm on the BJ wager ;-)
Hope this helps!
Dog Hand
