Blackjack is a card game that is
actually symmetric for the players and the house (dealer) and
one of few
casino games allowing an optimal play that reduces the house edge.
Obtaining such a strategy was not an easy task for the mathematicians
dealing with the game. The gaming events are formulated as referring to
sums of values and not specific combinations of cards of a
constant size. Any relevant probability for the gaming events in
blackjack is conditional: Calculating the probability that a player will
win the hand at a certain moment depends upon the future cards of the
dealer and changes with every card dealt. Moreover, this probability
alone cannot stand as the only criterion for a strategy, but only
compared with the conditional probabilities that the dealer will beat
the player’s hand or go bust.
Modern
blackjack is not very often played with one complete deck of cards as it
was in its early times; instead, it is played with a stack coming from
several decks (usually six), shuffled and divided off by a blank card at
about onefifth of the pack. This rule of course prevents knowing the
exact probabilities of occurrence of the various values of cards as the
cards are dealt, since they are not evenly distributed in the
composition of the stack in play.
For the games
played with complete decks of cards, we
first present the probabilities associated to card dealing and initial
predictions. The probabilities below are calculated for games
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 1deck game and C(104, 2)=5356 for 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. 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.
If
the game is played with complete decks of cards, the probability of
occurrence of a certain value during the game would be given by a simple
formula: If p(x) is the probability of a card with value
x being dealt as the next card, m is the number of decks
used, n(x) the number of xvalued cards already
dealt (seen), and N the total number of cards already dealt,
then:
P(x) = [4m
– n(x)]/(52m – N) if x is
different from 10 and
P(x) = [16m
– n(x)]/(52m – N) if x is 10.
Example
of application of the formula: Assume you play with one deck, you
are the only player at table, you hold Q, 2, 4, A (total
value 17) and the faceup card of the dealer is a 4. Let us calculate the
probability for you to achieve 21 points (receiving a 4) with the next
card dealt.
We
have m = 1, n(x) = 2,
N = 5, so:
.
For
the probability of achieving 20 points (receiving a 3), we have
n(x) = 0, N = 5, so:
.
For
the probability of achieving 19 points (receiving a 2), we have
n(x) = 1, N = 5, so:
.
If
we want to calculate the probability of achieving 19, 20 or 21 points, we
have to add together the three probabilities just calculated. We
get
P = 9/47 = 19.14893%.
When playing
with a random stack from several decks, such computation is not
possible, as we do not know the total number of cards of value x
are in the stack. In regard to developing an optimal strategy in
blackjack, this premise imposed a mathematical approach based on
approximations, through which to bypass the difficulty of probability
computations while not affecting in any way the optimal character of the
strategy. This was
the task of several mathematicians in the 1950s, whose work shaped the
first optimal strategy in blackjack.
The basic idea
was to assign constant probabilities to the individual card values,
namely 1/13 for the cards with value different from 10 and 4/13 for
those with value 10, regardless of the cards dealt in the progress of
the game. This approximation and supposition is of course adequate only
for a stack of cards of infinite size since the probabilities change as
the cards are dealt. However, the intended fixed strategy will
eventually approximate the optimal strategy on average.
Optimal
fixed strategy
The driving principle was that a good strategy should be oriented to the
dealer’s first card as carrying significant information about the course
that the dealer’s hand will likely take. The end probability
distribution for the dealer is shown in the next table:
Result: 17 18
19 20 21 blackjack bust
Probability: 0.145 0.139
0.133 0.180 0.072 0.047 0.281
The dealer
will go bust with a 28.1% probability, or in frequential terms, more
than one in every four hands on average.
A player who copies the house’s strategy, without
splitting or doubling, until at least 17 points are achieved will have
the same probabilities as those in the table above. If the game’s rules
were the same for the house as for the player, then the player’s
expectation for a bet would be zero. But this is not the case and a
player who follows the house’s strategy has actually an expectation of
minus 5.68% of the stake (a loss).
In
deciding how to play, a player should use the information given by the
dealer’s first card and compare the profit/loss expectations
(conditioned on the dealer’s first card and player’s current value in
hand) in the possible choices – drawing or not drawing another card,
doubling or not doubling, and splitting or not splitting. The higher
expectation indicates the optimal choice.
The applicable results of the
fixed optimal strategy are shown in the following tables of values of
the expectation, all of which have the dealer’s first card and player’s
total value as their rowcolumn variables (inputs).
The optimal
drawing strategy works in the reverse direction of the progress of the
game; that is, the player begins with the high totals and then optimizes
the strategy recursively step by step down to lower values with the aid
of the expectation tables.
Table1.
Probabilities for the dealer's results conditioned on the dealer's first
card.
Table 2.
Expectations, conditioned on the dealer's first card, for the player's
profit, when the player stays.
Table 3.
Expectations, conditioned on the dealer's first card, for the player's
profit, when the player draws.
Table 4.
Expectations, conditioned on the dealer's first card, for the player's
profit, when the player draws from a soft hand.
Table 5.
Expectations, conditioned on the dealer's first card, for the player's
profit, when the player plays optimally.
Table 6.
Expectations, conditioned on the dealer's first card, for the player's
profit, when the player plays optimally without doubling or splitting.
Table 7.
Expectations, conditioned on the dealer's first card, for the player's
profit, when the player draws exactly one more card.
Table 8.
Expectations, conditioned on the dealer's first card, for the player's
profit, for optimal play, depending on the splitting rule.
Table 9.
When splitting is advantageous; S indicates that splitting is
advantageous, while (S) indicates that splitting is advantageous only if
the next card is allowed to be doubled.
From the
tables one can see that such strategy is relatively defensive. For
instance:
– Against the dealer’s first card 4, 5, or 6, the player
should draw only to 11.
– Against the dealer’s first card 2 or 3, the player should
go to 13 and above.
– Against the dealer’s first card from 7 to ace, the player
should draw to 17.
Depending on the set of rules,
with the optimal play described above, the average loss can be brought
down between 0.64% and 0.88% of the initial wager (less than the average
loss for the simple bets in roulette, which is 1.35%).
The optimal
play based on comparison of mathematical expectations did not employ the
information available for the player about the cards already played
(viewed) and does not imply changes in strategy with the cards drawn,
but only has totals as inputs; this is why it is called the optimal
fixed strategy.
Highlowcount optimal strategy
In the 1960s
mathematician Edward Thorp investigated more extensively whether
the
information available about the cards already played
could significantly increase the player’s winning expectation and how it
could be quantified for use in an optimal strategy.
The basic idea
was to assign a weight to each card value played, as follows: +1 for 2
through 6; –1 for 10, face cards, and ace; 0 for the remaining cards.
This turned out to be a practical counting system based on an
approximation, whose total at a certain moment (called the count)
quantified sufficiently well the information given by the cards played
for being transformed into a mathematical criterion for changing the
strategy according to the count. It was called the highlow system.
The highlow counting system, although approximal in reflecting the
situation of the cards played, was proved by Thorp to reflect exactly
the change in winning expectation.
The
criterion driving the optimal strategy is the value of the fraction C/n,
where C is the result of the count and
n
the number of remaining cards in the deck/stack. The strategies of
drawing, doubling and splitting are expressed in tables of values for
100C/n, where the input variables are dealer’s first card
and player’s total in hand:
Bet: If 100C/n ≥ 3.6, then raise your bet;
otherwise, keep your bet at minimum.
Drawing:
Draw at "D" or when 100C/n is less than or equal to the given
value:
Table
10.
Draw at "D" or when 100C/n is less than or equal to the given
value:
Table 11.
Doubling:
Double
when
100C/n is greater than or equal to the given value:
Table 12.
Splitting:
Split on
"S" or when
100C/n is greater than or equal to the given value
(The table does not allow for split hands to be doubled):
Table 13.
Author 
The author of this page is Catalin Barboianu
(PhD). Catalin is a games mathematician and problem gambling researcher,
science writer and consultant for the mathematical aspects of gambling
for the gaming industry and problemgambling institutions.
Profiles:
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Researchgate 

