Duarte 1
Gavin Duarte
Math 20
Hamaker
30 November 2011
Blackjack stirs interest from all types of people, from the blackjack pioneering
mathematician Edward Thorpe to the vacationer traveling to Las Vegas for the weekend.1
Blackjack intrigues people because it is the only gambling game where the player actively
determines his or her odds, and one’s knowledge of strategy can greatly affect the probability of
winning. Because of the ability to alter the probabilities based on skill level, it is possible for an
experienced player to have a positive expected value of winning.
In order to fully comprehend the strategies behind blackjack, one must know the rules of
the game and how they affect play; in fact, most of the player’s options can be used to the
player’s advantage. The overall premise of the game is for the player to beat the dealer by
obtaining a sum of cards, a hand, less than or equal to 21 and at the same time have a higher
hand than the dealer. The game consists of one dealer and up to seven other players, and the
number of card decks, denoted by n, which ranges from n equals 1 to 6 held in a shoe. All
number cards equal their respective values, and each face card (jack, queen, king) is valued at ten
points. An ace can be worth one point, denoted a hard ace, and eleven, a soft ace, whichever
value makes the player’s hand better. Blackjack occurs when the first two cards dealt to a player
are an ace and a card valued at ten. Blackjacks are paid at 3/2 the original bet. The game starts
with the dealer dealing two cards to each player from left to right, the dealer last, and in turn,
1
Thorpe wrote Beat the Dealer: A Winning Strategy for the Game of Twenty-One (1962), the first attempt to
mathematically prove correct strategies for blackjack.
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each player can stay (keep the hand they have), or hit (draw another card) as many times as
desired. A player busts if the total is over 21, and automatically loses the hand to the dealer.
The rules for this game will be similar to those used in Las Vegas, having the following
criteria. A player initially bets a number B for certain hand, and may double down (place double
the original bet) upon receiving the first two cards. If the player receives a pair, they may split
the hand into two separate hands and place an additional bet, B. If the dealer’s up card (every
hand the dealer’s first card is placed face up) is an ace, the player can make an insurance side bet
of 𝐵⁄2 that the dealer gets a blackjack, and if this happens, the player gets B returned to him.
The rules for the dealer are very specific; they must continue to hit until they have a total of 17 or
greater, whether or not the dealer busts.
Why does the dealer have the edge?
The reason the dealer has the advantage is that they are always the last person to play, so
everyone at the table has a chance to bust (and therefore lose the hand) before the dealer has to
play.
The Basic Strategy
No matter what minor rules variations are present in each specific game of blackjack, it is
important that one remembers the basic strategies that help improve the player’s odds (Uston,
47).2 These points are often determined by the dealer’s upcard because that is the only piece of
information the player knows about the dealer, and the goal of blackjack is to beat the dealer.
1. If the dealer’s upcard is a 2 or 3, hit until you (the player) at least a total of 13. For
example, if the dealer had a 2 upcard, and your hand totaled 12, you would want to hit.
2. If the dealer’s upcard is a 4, 5, or 6, hit until you have at least a total of 12. The
reason the number total for the player is less than in part 1 is because the dealer has a
2
These strategies may be slightly different if one is using advanced counting techniques instead of a basic strategy.
Duarte 3
higher chance of busting given these upcards, because the dealer must hit until 17 is
reached.
For example, let 𝑋𝑖 denote the value of the 𝑖 𝑡ℎ card for the dealer. So 𝑋1 = 6 means the
dealer’s upcard is a 6.
10
𝑃(𝑑𝑒𝑎𝑙𝑒𝑟 𝑏𝑢𝑠𝑡𝑠 | 𝑋1 = 6) = ∑ 𝑃(𝑋2 = 𝑖) ∗ 𝑃(𝑋3 > 15 − 𝑖 | 𝑋2 = 𝑖)
𝑖=6
5
+ ∑ 𝑃(𝑋2 = 5) ∗ 𝑃(𝑋3 = 𝑖) ∗ 𝑃 (𝑋4 > 4 + 𝑖 | 𝑋2 = 5, 𝑋3 = 𝑖)
𝑖=1
5
+ ∑ 𝑃(𝑋2 = 4) ∗ 𝑃(𝑋3 = 1 + 𝑖) ∗ 𝑃 (𝑋4 > 4 + 𝑖 | 𝑋2 = 4, 𝑋3 = 1 + 𝑖)
𝑖=1
+ 𝑃(𝑑𝑒𝑎𝑙𝑒𝑟 𝑏𝑢𝑠𝑡𝑠 | 𝑋2 = 3) + 𝑃(𝑑𝑒𝑎𝑙𝑒𝑟 𝑏𝑢𝑠𝑡𝑠 | 𝑋2 = 2) + 𝑃(𝑑𝑒𝑎𝑙𝑒𝑟 𝑏𝑢𝑠𝑡𝑠 | 𝑋2 = 1)
= .42
So the dealer has a 42% chance of busting if their upcard is a 6, and this is why the
player needs only to total at least 12, because the probability of the dealer busting is
so high.
3. If the dealer’s upcard is anything 7 or higher (7, 8, 9, 10, or ace), hit until you have
at least 17. The reason is that the dealer has a higher probability of having a good hand
and not busting with this upcard, and if you have any hand lower than 17, the only way
you will win is if the dealer busts. Finding the probability of these situations is similar to
the way we found the probability in Part 2, where the dealer’s upcard is a 6.
4. Double down if you have a total of 11 and the dealer’s upcard is not an ace (i.e. 210). This is because the probability of you having a high hand is greater because you
Duarte 4
only need one ten valued card to reach 21, and even a card such as an 8 or 9 will still give
you a good hand.
5. Double down if you have a total of 10 and the dealer’s upcard is any card from 2
through 9. If the dealer and you both get a 10 valued card, you will still win the hand,
and if the dealer gets an ace, you will push (bet goes to the next round). That is why you
double down with a 10 or 11 only when your total is at least one number greater than the
dealer.
6. Split pairs when you have 8s or aces. The thought process behind this is that if you
split aces, you are turning a hand worth 12 into two very good hands both starting at 11.
If you have two 8s, 16 is a bad hand because you will only beat the dealer if he busts, and
instead you give yourself a better chance by starting off with two separate hands.
7. Never take insurance. If you have a blackjack, you might think of taking insurance so
that you will always win at least your original bet back. But this only occurs if the dealer
has a blackjack (natural) as well. You are better off taking the probability that the dealer
will not get blackjack, and then you will win 3/2 your original bet, because the
probability that the dealer has blackjack given that you have blackjack is smaller than the
probability that only you get blackjack (and therefore win 3/2 your original bet).
The Advantage of Card Counting
Many astute mathematicians have developed complex ways of card counting in order to
benefit the player’s odds. Card counting consists of making a specific value (point-count value)
for a certain type of card in each deck. This works because each card is a dependent trial,
therefore knowing more information about the previous trials will lead to a better understanding
of the current trial. First, we must define EOR, or the effect on the player’s expectation of
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removing individual cards. That is to say, when a specific card value is removed from the deck
(already played), the player’s expected value of winning changes. If the number is positive for
the respective card, it is advantageous to the player when this card is removed. Table 1 below
gives each EOR value by percentage change.3
Card
Value
EOR
2
3
4
5
6
7
8
9
10
Ace
0.381
0.434
0.568
0.727
0.412
0.282
-0.003
-0.173
-0.512
-0.579
Looking at these values, one can clearly see when cards 2-6 are removed, the players
expectation is significantly higher than the mean (µ=0), whereas if a ten or ace is removed, the
expectation is significantly lower than the mean. This can be interpreted in a more common
sense approach. If more low cards have been removed from the deck, and the dealer has a hand
of 12-16, there is now a greater probability that the dealer will draw a high card and bust. In
terms of different card counting techniques, one must determine the playing efficiency and
betting correlation of each technique, as well the overall ease of use. Betting correlation can be
computed by taking the dot product of the point count values and the EOR values, and then
dividing by the square root of the variance of the point count values and EOR values. This is
essentially the same as the standardized sum, because you are taking the values compared to
mean and dividing by the standard deviation. So, the betting correlation is
𝐵𝑒𝑡𝑡𝑖𝑛𝑔 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 =
𝛷
, where Φ is the
√(𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑝𝑜𝑖𝑛𝑡 𝑐𝑜𝑢𝑛𝑡 𝑣𝑎𝑙𝑢𝑒𝑠)∗(𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝐸𝑂𝑅 𝑣𝑎𝑙𝑢𝑒𝑠)
dot product of the point count values and EOR values (an example will be given later).4
The playing efficiency is computed by the quotient between the expected gain from the
lack of cards remaining in the deck and the hypothetical gain if the player had knowledge of all
Values obtained from Schlesinger’s Blackjack Attack
Note: The Theory of Gambling and Statistical Logic also gives equations for finding the SCORE, which combines
the values of betting correlation and playing efficiency.
3
4
Duarte 6
the cards. This is very similar to the cumulative distribution function that we learned in class,
except that it is the bivariate normal distribution. So, the expected gain, EG, without using a
basic strategy is
𝐸𝐺(𝑛) =
𝜎𝑛
2 ⁄2𝜎 2
𝑛
√2𝛱
𝑒 −µ
−
µ
√2𝛱
∞
∫
𝑒 −𝑥
2 ⁄2
𝑑𝑥
µ⁄𝜎𝑛
where µ is the expected gain using only basic strategy (see section The Basic Strategy), and
𝜎𝑛 is the standard deviation for the mean of an n-card subset of the deck, given by 𝜎𝑛 =
𝜎2 51−𝑛
√
𝑛
(
50
), which essentially accounts for the change in the standard deviation after a
certain number of cards, n, have been played. The hypothetical expected gain from using a
point count system is also bivariate normally distributed with correlation coefficient ρ.5
Therefore, the hypothetical expected gain,
𝐻𝐺(𝜌) =
𝜌𝜎𝑐
√2𝛱
𝑒
−µ2𝑐 ⁄2𝑟 2 𝜎𝑐2
−
µ𝑐
√2𝛱
∞
∫
𝑒 −𝑥
2 ⁄2
𝑑𝑥
µ𝑐 ⁄𝜌𝜎𝑐
where µ𝑐 and 𝜎𝑐 are the mean and standard deviation values using the point count system. So,
the playing efficiency is simply
𝐻𝐺(𝜌)
𝐸𝐺(𝑛)
The higher the betting correlation value, the lower the playing efficiency, because they
are inversely related, so if a certain card counting system has a high betting correlation, it will
most likely have a lower playing efficiency.
Hi-Lo Card Counting Strategy
5
Bivariate distribution functions have correlation coefficients defined under parameters defined at
http://mathworld.wolfram.com/BivariateNormalDistribution.html
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One card counting technique that is fairly easy to understand and implement is the Hi-Lo
system. This assigns a value of +1 for cards valued 2-6, 0 for cards 7-9, and -1 for a ten or ace.
Starting at zero, when a card is card is dealt the player simply keeps a running tally in their mind
about the total number by adding 1, 0, or -1 to the total. This system is considered to be
balanced since the total equals zero after all the cards are dealt because the expected value,
𝐸(𝑥) = 𝑛𝑝, 𝑠𝑜 𝐸(𝑥) = −1 ∗ 𝑃(𝑐𝑎𝑟𝑑 2 − 6) + 0 ∗ 𝑃(𝑐𝑎𝑟𝑑 7 − 9) + 1 ∗ 𝑃(𝑐𝑎𝑟𝑑 𝑡𝑒𝑛, 𝑎𝑐𝑒)
20
12
20
𝐸(𝑋) = −1 (52) + 0 (52) + 1 (52) = 0
Table 2 below illustrates the values for the cards.
Card
Value
Hi-Lo
Point
Value
2
3
4
5
6
7
8
9
10
Ace
+1
+1
+1
+1
+1
0
0
0
+1
+1
The betting correlation of the high low system would be the dot product of Table 1 and Table 2
divided by the product of the variances of table 1 and table 2. So,
𝐵𝑒𝑡𝑡𝑖𝑛𝑔 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 =
𝑇𝑎𝑏𝑙𝑒 1 ∙ 𝑇𝑎𝑏𝑙𝑒 2
√(𝜎 2 𝑜𝑓 𝑇𝑎𝑏𝑙𝑒 1) ∗ (𝜎 2 𝑜𝑓 𝑇𝑎𝑏𝑙𝑒 2)
=
5.15
√10 ∗ 2.842
= 0.966
because µ for both Table 1 and 2 equals zero.
The way the player uses this technique to his or her advantage is that if the total is greater
than 0, the player is at a slight advantage at that point in the deck, and the dealer has better odds
when the total is less than 0. The reasoning behind this is that if the Hi-Lo value is much greater
than 0 (e.g. 6), there is a much greater probability of being dealt a ten or ace, which allows the
player to have a stronger hand. Also, it makes the dealer bust more often because they must hit
on hands valued 12-16, which would therefore bust with a higher probability.
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Some downfalls of card counting are that a player will not be able to see every card being
dealt, and therefore the Hi-Lo value will never be exact as the decks diminishes. Also, the
advantage to the player can be lessened if a casino uses multiple decks of cards, uses burn cards
(cards placed face down before each hand), and reshuffling. The different tactics used by
casinos cause the player to have a disadvantage. Consequently, the fewer decks being use (1 is
ideal), and the more cards the player is able to see being played, the more effective card counting
becomes.
Other Ways of Modeling Blackjack- Markov Chains
Because blackjack uses cards without replacement, each subsequent card is dependent on
the cards played previously from the deck. This can make computations very difficult, and that
is why many of the values obtained for the game are approximated by doing a tedious amount of
simulations. However, if one thinks about blackjack in terms of moving from one value to the
next with a certain probability, from one state to another, Markov Chains can be very efficient at
modeling the complexities of blackjack.6
Because the dealer has specific rules to follow, a hand, H, which is 17 or greater, can be
thought of as an absorbing state, because the dealer must stand on those hands or have busted.
The state space representing the dealer consists of elements where
𝐻 = 2 − 11, where the first card is valued between 2 and 11
𝐻 = 4 − 16, where the first two cards total to a value between 4 and 16
𝐻 = 17 − 20, where the dealer must stand
𝐻 = 21, where the dealer has blackjack (only on the first two cards drawn
𝐻 > 21, where the dealer busts
6
A paper written by Rice University engineers called A Markov Chain Analysis of Blackjack Strategy
Duarte 9
(Note- Assume the dealer stands on soft 17)
Because the dealer must stand if 𝐻 > 16, the last 3 elements can be described as
absorbing states, because the probability of reaching that state again is 1. If one makes a
transition matrix, P, the dealer will reach an absorbing state in at most 17 steps (i.e. 𝑃17 ),
because the value must keep increasing until it reaches an absorbing state.
The same idea can be applied to the player’s hand, with the additional options of splitting
pairs, doubling down, and the ability to hit on any total less than 21. This leads to the player
reaching an absorbing state by at least 𝑃21 , because the player has more options (states) from
which to choose. The player’s expected return on a given hand can be determined by taking the
average of the player’s profit with both the player’s and dealer’s absorbing states given initial
card values.
Markov chains can also be used for card counting systems. For example, the Complete
Point-Count System uses an ordered triple, (Low, Medium, High), which has the same criteria as
Table 2. Using the ordered triple, the number of cards remaining in the shoe, R, can be
represented by 𝑅 = 52 ∗ 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑐𝑘𝑠 − 𝐿 − 𝑀 − 𝐻. This, in turn, leads to a formula
known as the High-Low Index (HLI)
𝐻𝐿𝐼 =
𝐿−𝐻
∗ 100
𝑅
This formula is much more useful than the Hi-Lo Technique for changing your betting amount.
Edward Thorp gives the following cases for betting given a uniform starting bet B and a given
HLI range:
[−100,2]
𝐻𝐿𝐼 = {(2,10]
(10,100]
𝑏𝑒𝑡 𝐵
𝐻𝐿𝐼
𝑏𝑒𝑡
𝐵
2
𝑏𝑒𝑡 5𝐵
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Common sense tells us that a high HLI favors the player (because there are more high
cards in the deck, the same as the Hi-Lo system) and therefore a player should bet more under
those conditions. This provides a slightly greater expectation for the player because they can
alter the betting amount as more cards in the deck become known. For example, if the dealer has
dealt 8 low cards, 4 medium cards, and 2 high cards in a one deck game, the
𝐻𝐿𝐼 =
8−2
∗ 100 = 15.8
38
so the player should place a bet of 5B for the next bet.
This leads to finding a distribution for a single hand, D, which is the total of all possible
distributions of each card type. So given a card value, for example a 2, would have the
distribution
𝑑(2) =
1 20𝑁 − 𝐿
∗
5
𝑅
where N= number of decks. We multiply by 1/5 because 2 is one of 5 low valued cards, and
there are 20 low valued cards per deck. The same approach can be applied to both the medium
and high valued cards. Using the distributions for the specific card values, one can average the
triples and create the probability of the player’s advantage using Markov chains by starting at a
specific state in the transition matrix (depending on the dealer’s upcard and player’s hand) and
then determining the probabilities of each specific outcome (bust, win, etc.).
Conclusion
A small change in the rules of blackjack can have a profound effect on the expected
winnings for the player. However, learning the basic strategy behind blackjack allows anyone to
better themselves without much prior knowledge about math. However, the dealer will continue
to have an advantage unless the player uses more advanced techniques. If the player uses card
counting techniques, the expected profit can reach +0.02 if using only one deck, where 0 is
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breaking even.7 However, the more decks that are in the shoe, the less the player has an
advantage. A player with no real experience has an expected loss of -0.08, while the basic
strategy will get you to around -0.01 to -0.03. Clearly, the percentage changes are small, and that
is why these techniques must be implemented for long periods of time in order for them to be
useful. Even when using the more exact Markov Chains to find the probabilities, it still only
yields an expected gain of +0.02. Even though some basic assumptions are made to simplify the
computations, the strategies still proved to be very accurate to the actual probabilities. Because
each successive hand is dependent on the previous hand, finding the exact probabilities can get
rather messy (as seen in Basic Strategy, Part 2). This is why many of the numbers are found
using simulation over hundreds of thousands of times. Using the player options such as splitting
and doubling down is the only way for the player to create an advantage, and the player must
rely on their own intuition to make these decisions. However, blackjack remains the only casino
game where the player can have the advantage, and models such as Markov Chains help to
illustrate the math behind the game.
7
Formulated in Million Dollar Blackjack via numerous simulations
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Appendix
The probability of the dealer busting given the dealer’s upcard value.8 This correlates with the
section Basic Strategy, Part 2
Dealer’s
2
3
4
5
6
7
8
9
10
Ace
Upcard
P(Dealer
0.353 0.3756 0.4028 0.4289 0.4208 0.2599 0.2386 0.2334 0.2143 0.1165
Busts)
The probability of busting while taking a hit. This is fairly straightforward. For example, if you
have a hand equaling 20, the only way of not busting is if you are dealt an ace, so
𝑃(𝑏𝑢𝑠𝑡 | 𝑡𝑜𝑡𝑎𝑙 = 20) = 1 − 𝑃(𝑛𝑜 𝑏𝑢𝑠𝑡 | 𝑡𝑜𝑡𝑎𝑙 = 20) = 1 − 4/52 = 0.92
Hand
Total
P(Busting)
<11
12
13
14
15
16
17
18
19
20
21
0
0.31
0.39
0.56
0.58
0.62
0.69
0.77
0.85
0.92
1
The overall expected profit of the player if using the Complete Point Count System (in the
Markov Chain section) in terms of increasing deck sizes9
Number of Decks
E(Profit)
8
9
1
0.0296
2
0.0129
4
0.003
Table from http://www.blackjacktactics.com/blackjack/odds
Table gathered from results of Markov Chain Analysis in A Markov Chain Analysis of Blackjack Strategy
Duarte 13
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