We
first present the probabilities attached to card dealing and initial
predictions. In making this calculus, circumstantial information such as
fraudulent dealing is not taken into account (as in all situations
corresponding to card games). All probabilities are calculated for cases
using one or two decks of cards. Let us look at the probabilities for a
favorable initial hand (the first two cards dealt) to be achieved. The
total number of possible combinations for each of the two cards is C(52,
2) = 1326, for the 1deck game and C(104, 2)=5356 for the 2deck game.
Probability of obtaining a natural
blackjack is P
= 8/663 = 1.20663% in
the case of a 1deck game and P
= 16/1339 = 1.19492% in
the case of a 2deck game.
Probability
of obtaining a blackjack from the first two cards is P = 32/663 = 4.82654% in the case of a 1deck game and
P = 64/1339= 4.77968% in
the case of a 2deck game.
Similarly,
we can calculate the following probabilities:
Probability
of obtaining 20 points from the first two cards is P = 68/663 = 10.25641% in
the case of a 1deck game and P
= 140/1339 = 10.45556% in
the case of a 2deck game.
Probability
of obtaining 19 points from the first two cards is P = 40/663 = 6.03318%
in the case of a 1deck game and
P = 80/1339 = 5.97460%
in the case of a 2deck game.
Probability
of obtaining 18 points from the first two cards is P = 43/663 = 6.48567%
in the case of a 1deck game and
P = 87/1339 = 6.4973% in
the case of a 2deck game.
Probability
of getting 17 points from the first two cards is P = 16/221 = 7.23981%
in the case of a 1deck game and
P = 96/1339 = 7.16952%
in the case of a 2deck game.
A
good initial hand (which you can stay with) could be a blackjack or
a hand of 20, 19 or 18 points. The probability of obtaining such a hand is
calculated by totaling the corresponding probabilities calculated
above: P = 32/663 + 68/663 + 40/663 + 43/663 = 183/663, in the case
of a 1deck game and P = 64/1339 + 140/1339 + 80/1339 + 87/1339 =
371/1339, in the case of a 2deck game.
Probability
of obtaining a good initial hand is
P = 183/663 = 27.60180%
in the case of a 1deck game and
P = 371/1339 = 27.70724%
in the case of a 2deck game.
The
probabilities of events predicted during the game are calculated on the
basis of the played cards (the cards showing) from a certain moment. This
requires counting certain favorable cards showing for the dealer and for
the other players, as well as in your own hand. Any
blackjack strategy is
based on counting the cards played. Unlike a baccarat game, where a
maximum of three cards are played for each player,
at blackjack many cards
could be played at a certain moment, especially when many players are at
the table. Thus, both following and memorizing certain cards require some
ability and prior training on the player’s part.
The
formula of probability for obtaining a certain favorable value is similar
to that for baccarat and depends on the number of decks of cards used. If
we denote by x a favorable value, by nx the number of cards
showing with the value x (from your hand, the hands of the other
players and the face up card in the dealer’s hand) and by nv the
total number of cards showing, then the probability of the next card from
the deck (the one you receive if you ask for an additional card) having
the value x is:
This
formula holds for the case of a 1deck game. In the case of a 2deck game,
the probability is:
Generally
speaking, if playing with m decks, the probability of obtaining a
card with the value x is:
Example
of application of the formula: Assume play with one deck, you
are the only player at table, you hold Q, 2, 4, A (total
value 17) and the face up card of the dealer is a 4. Let us calculate the
probability of achieving 21 points (receiving a 4).
We
have nx = 2, nv = 5, so:
.
For
the probability of achieving 20 points (receiving a 3), we have
nx = 0, nv = 5, so:
.
For
the probability of achieving 19 points (receiving a 2), we have
nx = 1, nv = 5, so:
.
If
we want to calculate the probability of achieving 19, 20 or 21 points, all
we must do is total the three probabilities just calculated. We obtain
P = 9/47 = 19.14893%.
Unlike
in baccarat, where fewer cards are played, the number of players is
constant (two), and the number of gaming situations is very limited, in blackjack, the number of possible playing configurations is in the
thousands and, as a practical matter, cannot be entirely covered by tables
of values.
Sources 
A big part of the gaming situations that require a decision, where the
total value held is 15, 16, 17, 18, 19 or 20 points, is comprised in
tables in the section titled Blackjack of the book PROBABILITY
GUIDE TO GAMBLING: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack,
Poker, Lottery and Sport Bets.
You will also find there other issues of probabilitybased
blackjack strategy . See the Books
section for details. 

