INDEX

1. ABSTRACT

2. INTRODUCTION

3. RESULT AND ANALYSIS OF GAME

4. UML DIAGRAMS FOR THE GAME

5. SCREEN SHOTS

6. CONCLUSION

7. ACKNOWLEDGEMENT

8. FUTURE WORK

9. REFERENCES

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Abstract: Blackjack is the most popular table game in casinos because it is one of few games where the object is to beat the dealer. This type of game allows the player to use certain skills to guide the success or failure of a player against the dealer. With basic strategy or zero-memory strategy, players are able to use the proper skill and correct rules when faced with decisions on hitting, standing, double down, or splitting against the dealer’s known up card. The calculations of the expectations and probabilities under zero-memory strategy are categorized into Tables. Using these Tables, along with various betting techniques such as card counting, random betting, and exponential betting, will increase a player’s maximum expectation and give them an advantage over the dealer. In order to test these strategies and techniques, a total of 240 trials were conducted while applying the three betting techniques to each trial. A computer game, Hoyle Casino99’, was used to carryout the basic hi/lo card-counting technique that is applied in my project. The results will include tables from the multiple trials of the various betting techniques vs. 1, 2, 3, or 4 players. The output of statistical data will show an increase in the player’s “win probability” using the basic strategy technique and will also show which betting technique is the best for increasing a player’s total amount of chips won.

Introduction :

Blackjack, considered to be the most popular table game in casinos, rakes in

about 15% of all casino revenue . It is the only casino game where the player

can have an “edge” over the casino. In order to have that edge over the

casino, a player’s rate of profit on his investment should be greater than the

casinos. The only actually set “house edge” over the player is if the player

and dealer both bust, then the dealer wins. Typically the “house edge” over a

player is dependent on the player’s decision so it is very difficult to say

exactly what the house edge is over the player. With a beginning blackjack

player, the house edge against the player could be anywhere from two to three

percent. A more advanced player who is usually a player that follows a basic

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set of rules of when to hit, stand, double, could possibly lower that edge to

about ≤0.5%.

1.1 The Game of Blackjack

The rules of blackjack are quite simple. The object of blackjack is to beat the

dealer’s cards totaling 21 or less and also having a higher total number of

points than the dealer. A player can also win if the dealer’s cards exceed 21

and the player’s cards do not. Exceeding a total of 21 is known as “busting.”

The casino’s advantage to blackjack is when the players go first in deciding

what to do with their cards. If the player busts, they automatically lose even if

the dealer busts. The value of each card is straightforward. All cards count

their face value in blackjack. For example, a 2 of hearts is worth 2 points, a

ten of diamonds is worth 10 points, etc…. Face cards are also worth ten

points and an ace is counted as either 1 point or eleven points depending on

the total value of the hand. It is usually assumed that aces are worth 11

points, but if a player’s total exceeds 21, it will be counted as 1 point. The

total of any hand is the sum of the card values in the hand. For example, if

you are dealt 8, 3, 5, and 4, your total is 20. There are essentially two types of

hands, hard and soft. A hard hand is defined as one with no aces. For

example, 10 and 8 is a hard hand. A soft hand is one with an ace in the hand.

For example, Ace, 7 is a soft hand of 18 or “soft 18.” Prior to the deal of the

cards, all players must make a bet by placing chips in their respective betting

boxes.

Every player and the dealer will receive two cards. One of the dealer's card

(known as the dealer's up card or face card) is dealt up so that players can see

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its value. The other dealer's card (known as the dealer's down card or hole

card) is unseen. The two player cards can be either dealt face up, face down,

or sometimes one up and one down. After player has received two cards from

the dealer, the player has the option of hitting, standing, splitting, or doubling

down. A hit, means that the player would like the dealer to give you another

card to your hand. Standing means you are satisfied with the total of the hand

and want to stand with the cards you have. You can only split your cards if

you have two “like” cards (e.g. a pair of 4's or jacks). When you split you

must make another bet equal to your original bet by placing your chip next to

the original chip bet on the hand. When pair splitting, a player will play each

card as a separate hand and can draw as many cards as a player would like to

each hand (except split aces-most casinos will only allow one draw card to

each ace). For example, if a player were dealt a pair of 8's (16) and split, a

player would have two separate hands containing an 8. One would be

required to play out one of the split hands first before the other. Most casinos

will also allow players to split all 10 value cards such as a jack and ten or

queen and king. If a player chooses to double down, he or she doubles the bet

in return for receiving one and only one draw card. In most casinos you can’t

double down after hitting. Other options such as surrender and insurance are

also used in blackjack. However, in this study, we will not be dealing with

insurance or the surrender option.

Unlike players, the dealer in blackjack has no playing option. Casino rules

specify that a dealer must draw when the dealer's hand totals less than 17 and

stand when the total is 17 to 21. In some casinos, dealers must stand on soft

17 and in others they must hit (it's better for the player if the rules specify the

dealer must stand on soft 17). “Multiple deck” blackjack or 4-6 deck

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blackjack is most commonly seen in casinos. However, there are such casinos

that have one deck blackjack, but in our case we will be dealing with 6-decks.

[2] [4] [6]

.

1.2 Player’s Strategy

If the dealer has a six showing and the player has a card value total of 12, do

you hit or should you stay, hoping that the dealer busts? Unless you are card

counting, this is an example of zero-memory strategy, meaning the only

known cards are the dealer’s up card and the players two up cards. Under the

zero-memory strategy of one deck of cards, there are a possible 55 two-card

hands a player can receive off the deal. For example, a player may be dealt a

(9, 10), (7, 4), (K, 3) etc. Knowing that only 2 cards are dealt to the player

and there are 10 different point values for each card (since 10’s, J’s, Q’s, and

K’s are all worth 10), then the total amount of two-card hands can be found

using combination. In our case, we take “ten choose two” for the amount of

two-card combinations for the 10 different point values of the 52-card deck.

We also add, “ten choose one” for the amount of two-card combinations that

have the same point value, such as a (2,2), (10,10), (7,7) etc., where (10,10) is

considered to be pairs of face cards also. The equation for this can be shown

below:

10C2 + 10C1 = 45 + 10 = 55 possible two-card hands

The only card that is known of the dealers is the up card for a total of 10

possibilities of values such as 2-9, 10 (10, J, Q, K), and 1 or 11 (A). The

player’s zero-memory strategy then must be defined for 550 possible cases or

55 two-card hand possibilities times the 10 possibilities of the dealer’s up-card

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(55x10 = 550). Under zero-memory, we can produce probabilities of the

player’s mathematical expectation of hitting, standing, doubling down, or

splitting. Defining these probabilities from [4]:

Let random variable t0 be the two-card total dealt to the player and let k be the

final total of the dealer’s. Then if the player decides to stay or “stand” with

his two-card total of t0, the player’s single-play probability of winning Pw,

would be defined as the probability that the player’s two-card total, t0, is less

than or equal to 21 and greater than the dealers final total, k, or the probability

that the dealers final total cards, k, is greater than 21 and the player’s two-card

total, t0, is less than or equal to 21. The probability of losing is then defined as

the probability that the player’s two-card total, t0, is less than the dealer’s final

total, k, which is less than or equal to 21. The probability of tying, or

“pushing”, is then the probability that the dealer’s final total, k, is equal to the

player’s two-card total, t0. All of which are shown below:

Pw = P(k < t0 ≤ 21) + (P(k > 21)P(t0 ≤ 21))

PT = P(t0 = k)

PL = P(t0 < k ≤ 21) where Pw + PT + PL = 1

Defining a player’s mathematical expectation E(X): if a random variable X

can assume any n values, x1, x2, … xn with respective probabilities p1, p2, …, pn,

then E(X) is expressed as:

E(X) = p1 x1 + p2 x2 + … + pn xn = ∑ pi xi, where i goes from 1 to n,

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Using this definition, we can find a player’s mathematical expectation E(t0) of

standing with a their two-card total t0 for a n dollar amount bet for each hand,

where n = 1, 2,…, d dollar amounts:

E(t0) = n(Pw) – n(PL) = 2n(Pw) + n(PT) – n = 2n(P(t0 > k) + P(k > 21)) +

n(P(to = k))

Let random variable t1 be the player’s total card value of the player’s after

drawing one card or “hitting.” Then the probability of winning is defined as

the probability of the players total card value, t1, less than or equal to 21 and

greater than the dealer’s final total card value, k and the probability that the

dealer’s final total, k, is greater than 21 and the player’s total card value, t1, is

less than or equal to 21. Also, the player’s probability of losing is defined as

the probability that the player’s total card value, t1, is less than the final total

of the dealer cards, k, less than or equal to 21 and the probability that the

player’s total card value, t1, is greater than 21. Then the probability of player

tying the dealer is the probability that the player’s total card value, t1, is equal

to the final total of the dealer’s cards, k, and less than or equal to 21. The

player’s single-play probability of winning Pw, of tying PT, and of losing PL

are shown as:

Pw = P(k < t0 ≤ 21) + P(k > 21)P(t1 ≤ 21)

PT = P(t1 = k ≤ 21)

PL = P(t1 < k ≤ 21)+ P(t1 > 21)

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Using the definition from above, the player’s mathematical expectation E(t1)

can then be defined as the probability of a player who draw’s exactly one card

obtaining a total three card value of t1 for n betting amount is:

E(t1) = n(Pw) – n(PL) = 2n(Pw) + n(PT) – n =

= 2n(P(k < t0 ≤ 21))+ 2n(P(k > 21)P(t1 ≤ 21)) + nP(t1 = k ≤ 21)

– n

Then for every value t0 the player has the option of staying or drawing to

increase the total of t1. It is the player’s advantage to “hit” if E(t1) > E(t0) and

it is the player advantage to “stand” if E(t1) < E(t0), so taking the difference of

the two:

E(t1) - E(t0) = 2nP(k < t0 ≤ 21) – 2nP(t0 > k) – 2nP(t1 > 21) P(k > 21)

+ nP(t1 = k ≤ 21) – nP(t0 = k)

Eventually if the player keeps drawing and doesn’t decide to “stay”, E(ti+1) -

E(ti) < 0, drawing an additional card (t2) will change the new total hand value

until the maximum value of ti produces the state E(ti+1) - E(ti) ≥ 0 for which the

player then stays on his total of ti ≤ 21.

For example, suppose a player is dealt a 7 of diamonds, 5 of hearts for a total

card value of 12. The dealer’s up card shows a 6 of hearts. If the player’s bet

was n = 1, then the player’s mathematical expectations of hitting E(t1) and

standing E(t0) are:

E(t0=12) = 1Pw – 1PL = 2Pw + 1PT – 1 = 2P(12 > k) + 2P(k > 21) + 1P(12 = k)

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E(t1) = 2Pw + 1PT – 1 = 2P(k < t0 ≤ 21) + 2P(k > 21)P(t1 ≤ 21) + 1P(t1 = k ≤ 21) – 1,

Knowing the dealer’s up card, 6, the probability of the player’s card total of

12 greater than the dealer’s final total k, P(t0 > k), is the possibility of the

dealer’s down card being a 5, 4, 3, 2, and A if dealer draws on soft 17. Then

the P(k > 21) is the probability of the dealer drawing until they “bust” and the

P(12 = k) is the probability of the dealer’s down card being a 6 and the

probability of the dealer drawing until to = k. Similarly, one can calculate the

player’s mathematical expectation of hitting E(t1). Then taking the difference,

E(t1) - E(t0) = -0.0230, E(t1) < E(t0) which means the player should stand in

this situation. So if a player is dealt a two-card value of 12, no matter what

cards, if the dealer’s up card is 6, the player has more of an advantage of

staying in this situation, thus relying on the dealer drawing until they have a

total, k > 21, or busting. Table A shows the player’s mathematical

expectation of standing, hitting, or doubling down under the zero-memory

strategy for player’s two-card total of 11, 12, and 13, vs. the dealer’s up-cards

of 5 through 9 found in [4]. The full table can be found in the Appendix,

Table I.

TABLE A: A player’s expected gain for 4-deck blackjack

Player’s

two-card

hand

Dealer’s Up-card

5 6 7 8

9

13 -.1597 -.1547 -.2997 X -.3506 X -.4077 X

12 -.1578 -.1533 -.2203 X -.2807 X -.3438 X

11 .6401 D .6814 D .4674 D .3485 D .2268 D

* Conditioned on No dealer or player blackjacks

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* Stand unless: X = Hit/Draw, D = Double-Down

Looking at the table, one can see why we should stay on a 12, 13, with a

dealer’s up-card of a 5 or a 6 because the probability of the dealer having a

face card as the down-card for a total of a 15 or a 16, and the dealer hitting a

card value of a 7 or greater (busting), is greater than the player’s probability of

hitting and receiving a 10 or a face card and busting.

This zero-memory strategy can be summarized into more simple form known

as basic strategy. The basic strategy technique offers the same insight, on

whether to hit, stay, double down, or split, as the zero-memory strategy for 4-

decks. Basic strategy takes in account for the dealer’s up-cards and the

player’s two-card total as in the zero-memory strategy. It will also maximize

a player’s expectation, bringing them closer to even with the house. Table B

shows this basic strategy for multiple decks (4-6 decks) for player’s two-card

total of 10 through 13, vs. dealers up cards of 2 through 7. This table was

used for carrying out the various trials and hands in the study. The full

version can be found in the Appendix as Table II.

Table B: Basic Strategy for Multiple Decks

Player’s

two-card

hand

Dealer’s Up-card

2 3 4 5 6

7

13 Stand Stand Stand Stand Stand Hit

12 Hit Hit Stand Stand Stand Hit

11 Double Double Double Double Double Double

10 Double Double Double Double Double Double

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Using the basic strategy or zero-memory strategy, a player will be able to

determine the correct way to play his or her hand knowing the dealer’s up-

card. Along with various betting techniques such as, card counting, random

betting, and exponential betting, we can examine which technique yields the

higher win probability for each case vs. the dealer and also the total amount

won for the game of Blackjack. [2] [3] [4] [6]

1.3 Background and Motivation

Zero-memory strategy is directly related to probability theory and statistical

analysis, which are a part of counting process in combinatorics. Zero memory

strategy deals with probabilities and expectations of a player hitting, standing,

doubling down, or splitting vs. the dealer’s known up card. Using the zero-

memory technique alone can increase your winning percentage and give you

and edge over the casino. Applying the basic strategy to other betting

strategies such as, card counting, random, and exponential betting techniques,

will help increase your winnings and will maximize total winning amounts

even more. This project will focus on increasing a player’s expected gain in

winnings by using the strategy applied with card counting and various betting

techniques. The concept of applying zero-memory strategy or basic strategy

to betting techniques has applications to real world situations such as playing

blackjack in casinos, Random Variables in probability, and in math/statistic

field.

1.4 Known Work in Math Literature

Similar research has been done using the basic strategy applied with a card-

counting betting technique. Millman [5] used a Vax 11/780 computer to

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determine what advantage a player using zero-memory strategy might obtain

over the house. It computed a table, just like in [2], on whether a player

should hit, stay, double down, or split knowing the dealer’s up card and the

player’s two-card total only. Using this basic strategy table and the hi/lo card-

counting betting technique, they were able to determine if the technique

increased a player’s proportion of wins (win probability), the mean total of

chips won, and the mean edge over the dealer or house. The simulation was

performed under Vegas and Atlantic City blackjack rules with four decks.

Data was collected from 600 trials and 400 trials of twenty thousand hands in

both Las Vegas and Atlantic City respectively. The results demonstrated that

Las Vegas is more desirable from the player’s point of view than Atlantic

City. The win probability turned out to be 87% for Las Vegas and 79% for

Atlantic City. The average chip amount won for trials in Las Vegas was 2084

and in Atlantic City, 1108, which is almost double the amount. The mean

Player “edge” was 1.35 and .91 for Las Vegas and Atlantic City respectively.

Some of the other zero-memory strategy in Blackjack information was taken

from [1] and [4]. Much of the information that for this research has come

from [2], [4], and [5]. Since this project deals with probabilities and

expectations, research was done by looking at many optimal strategies and the

mathematics behind these probabilities, which were found in [1] and [2].

2 Proposed Method

The zero-memory strategy or basic strategy sets rules for us players about

when to hit, stay, double down, or split. Using the zero-memory strategy, a

player will be able to determine the correct way to play their hand knowing

only the dealer’s known up card. In order to carry out the trials for card

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counting, a computer program called Hoyle Casino 99’ was used. The Hoyle

Casino 99’ computer game allows you to play a variety of casino gambling

games such as blackjack, poker, slots, keno, craps, and roulette. Players may

start a new profile with a starting bankroll of five thousand dollars and can

increase their bankroll by playing games and gambling more money. The

maximum amount of money you can earn in your bankroll is unlimited.

However, if you end up losing the entire $5,000, you may reset your

character’s profile and start with another five thousand. Blackjack is just one

of the many casino games that you can play. The tables range from $5-

$1,000, $10-$2,000, $25-$3,000, and $100-Unlimited, minimum bet-

maximum bet respectively. In our case, we will be playing on the $100-

Unlimited table since our initial and minimum bet is $100 for each betting

technique. This game allows me to set up how ever many decks I want, (I our

case, we will use 6-decks), how many players there are, and also the different

rules applied to the game such as; re-splitting, doubling down after splitting,

dealer hitting on soft 17, etc… It also has a card-counting feature that keeps a

running count of the cards that come up in the hands played. The chip are

divided up into amounts $100, $500, $1K, $5K, $10K, etc… which allowed

me to set the bets accordingly to the running count in multiples of 100. This

game is very useful and quick in getting information on the card-counting

method and how it is applied with the basic strategy technique.

For the exponential and random betting techniques, the trials and hands were

carried out by hand using 6 decks of cards, and recording each hand for each

of the 20 trials of the four cases vs. the dealer following the basic strategy

table from [2] and [6]. The data that was recorded on the sheets was the card

amounts of the player and dealer, whether or not the player hits, stays, splits,

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or doubles down, and whether or not the player wins or loses the hand. After

each case, the various bets were applied to each of the hands and the total

number of wins, losses, and the total amount won and lost were recorded on

the sheets as well. Overall, there are 20 trials for each case of 1-4 players vs.

the dealer, applied to each of the three betting techniques totaling 4,800

recorded hands. All recorded results are displayed in the Appendix. These

results will show us which betting technique yields a greater maximum

expectation using the basic strategy and how effective the basic strategy

technique is in yielding a higher percentage of wins overall.

2.1 Random Betting Technique

One of the betting techniques applied to basic strategy or zero-memory

strategy in this project is Random Betting, where a betting amount for each

hand is randomly generated by a simple random sample. Since all trials

consisted of exactly 20 hands, a random sample of 20 numbers, listed one

through twenty, are taken and applied to each hand in each trial. In order to

get a random sample of 20 numbers, we will use the Simple Random Sample

Generator, a statistical applet found online. [7] This applet allowed me to

carry out 80 lists of 20 random numbers. For example, one list may be, 17, 1,

9, 3, 16, 8, 4, 5,…,13, 20, 14, 12. Taking these numbers and multiplying them

by 100, we will get a new list of numbers in multiples of 100, starting with

100 as our smallest possible bet and 2000 being our biggest possible bet. So

in our previous list, our first bet on hand 1 would be $1700, then $100 on

hand 2, then $900 on hand 3, and so on.

2.2 Exponential Betting Technique

Another technique applied to basic strategy is Exponential Betting, whereby

the amount of bet on one hand is doubled of that on the previous hand until

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the desired amount is attained. For example, at a blackjack table with a

minimum bet of $100, our first initial bet would be $100. Then our 2nd hand

would be $200, and doubling it another time to get our 3rd hand bet of $400,

then $800, $1600, $3200, $6400, etc… Since all trials consist of twenty

hands, on the twentieth hand, according to the definition of exponential

betting, you will place a bet of $52,428,800, which is a lot of money! Using

this technique over 20 hands can either make a player very rich or will make a

player far in debt depending on the hand amount and what it is worth. That is

why this study is assuming that the initial bankroll is unlimited and that

exponential betting is to be defined as the technique of doubling bets until the

player attains a certain amount over the 20 hands. In this study, the desired

amount is $50,000 or more.

2.3 Card Counting Technique

Using the basic Hi/Lo card counting system, every card is assigned a number

value: 2-6 = +1, 7-9 = 0, 10, J, Q, K, A = -1 and for every card that comes up,

each value of each card is kept track of in a running count (see [6]). The

betting technique applied to the card counting system is to bet accordingly to

your running count. For instance, if a player were at a blackjack table with a

minimum bet of $100 and our running count is at a +5, you would bet 5 times

the initial betting amount or minimum bet at the table, or $500 dollars for the

next hand. If the running count is at a +1 or less, you would bet the minimum

bet of $100. Player’s must keep a running count of the cards that come up

during play. For example, if the last round of cards dealt were K, 10, 8, 6, J,

3, 2, A, 5, 5, 7, a player’s running count would be (-1) + (-1) + (0) + (+1) + (-

1) + (+1) + (+1) + (-1) + (+1) + (+1) + (0) = +1 and so our next bet for the

next hand would be $100.

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3 Results and Analysis of Table C

Over a period of 2 months, four thousand eight hundred blackjack hands were investigated in which the player employs our basic strategy technique. Data was collected for samples consisting of 80 trials for each case of 1, 2, 3, and 4 players vs. dealer. Each sample was applied to the three betting techniques for a total of 240 trials, each trial consisting of 20 hands. All cases for number of players vs. dealer were investigated for each of the betting techniques that were applied. Let W be the number of wins and L be the number of losses. Then the probability that a player’s final bankroll exceeds his initial bankroll after a trial, denoted WP, will be called the player’s win probability. Likewise, we call the player’s loss probability, denoted LP, just 1 - WP. Table C gives a summary of the sample data of the player’s number of wins, losses, the win probability, and loss probability for each betting technique, basic strategy in general, and the overall total for all 4,800 hands. The Basic and Overall rows in Table C define wins and losses as whether or not the player wins or losses the hand vs. the dealer. Therefore a win, in this case, does not depend on the final amount after each trial but depends on the outcome of after each hand. For each of the betting techniques, the number of wins and losses are defined as whether or not the player came out ahead after 20 hands. The results of Table C are shown below: TABLE C:

(# of Players)

W=Wins 4 3 2 1

L=Losses4 3 2 1

WP

4 3 2 1

LP

4 3 2 1

Exponential

13

17

15

5 7 3 5 15

.65

.85

.75

.25

.35

.15

.25

.75

Random

17

16

16

14

3 4 4 6 .85

.80

.80

.70

.15

.20

.20

.30

Hi/Lo 12

5 14

12

8 15

6 8 .60

.25

.70

.60

.40

.75

.30

.40

16

Basic 662

696

698

668

538

504

502

532

.552

.580

.581

.557

.448

.420

.419

.443

Overall 2724 2076 .5675 .4325

W: the number of wins by 1 player vs. the dealerL: the number of losses by 1 player vs. the dealer

WP: the probability of 1 person winning vs. the dealerLP: the probability of 1 person losing vs. the dealer

Each column is divided into four sections for each of the four cases of 1, 2, 3, 4 players vs. the dealer. The first column shows the total number of wins over each trial, the second column shows the total number of losses over each trial, the third column shows the player’s win probability, and the fourth column shows the loss probability. It clearly shows that there is a higher percentage of wins overall while applying the basic strategy technique. This may mean that by using the basic strategy technique, a player has the advantage over the casino. Knowing this, a player can use this technique to their advantage to increase their winning percentage, which can help them to come up ahead in their overall winning amount as seen in [4] and in [5]. In [4], they were able to use the basic strategy technique or zero-memory technique along with the hi-lo system to find out that it was sufficient in achieving a positive expected gain in the casinos that were tried. And Millman [5], shows that by using this basic strategy technique, they were able to yield a positive mean winning amount in both Las Vegas and in Atlantic City. The proportions of wins were significantly higher in both places as well.

The number of players at the table does not affect a player’s chances of winning a certain hand under the random betting and exponential betting technique. The only effect it has on these two betting strategies is the overall amount won in the end. The more players that are at the table means that there are more cards being dealt out of the deck and distributed for each hand vs. the dealer. However, in card counting, the number of player’s at the table has a different effect on the outcome of your wins. In card counting, the running count may vary dramatically accordingly to the cards that are dealt out of the deck for each hand. For example, there may be a string low card’s dealt out in one hand vs. the dealer, making the running count for the next hand significantly positive, causing the player to bet a large amount on the

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next hand. With this in mind, we can carefully examine the results of Table C. There does seem to be somewhat of a relationship between the amounts of wins at the table with more players than fewer players under the random betting technique. Looking at column 3, in the case where there are four players vs. dealer, 85% of the time the player will end up with an amount greater than the initial amount after 20 hands, i.e. player will have a positive expected gain. In the cases for 3, 2, and 1 player vs. the dealer, the percentage of the amount of wins is 80%, 80%, and 75% respectively. However, it isn’t obvious of a relationship for the exponential and card-counting betting technique because the results for the wins and losses vary between the numbers of players at the table and are not as consistent as for the Random betting technique. Of course many more trials and hands with more than four players should be carried out in order to safely conclude there is a relationship, but from the results, a player can use the random betting technique to their advantage and sit at a table with more players around it to walk out of the casino ahead.

3.1 Results and Analysis of Table D

Next, let us also define the player’s edge in 1600 trials to be the net number of wins over the net number of losses, denoted e. Table D reports the player’s “edge” over each case.

TABLE D:

Cases 1 player 2 players 3 players 4 players Overall

Edge (e) 1.230 1.381 1.390 1.256 1.312

Edge (e) = the number of times a player wins / the number of times a player loses

Table D shows that there is an overall positive player’s “edge,” meaning the total amount won over the total amount loss is greater than 1. The total average player’s “edge” over 4,800 hands is 1.312, which is also greater than 1. This means that there are a higher number of wins vs. losses; meaning higher percentage of wins overall using this basic strategy technique. Millman [5] also found the “mean player edge” for both Las Vegas and Atlantic city to be greater than one. The mean player edge in Las Vegas was 1.35 and the overall mean edge was 1.13, which is also greater than one,

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meaning there is a greater amount of wins than losses while using the basic strategy technique. Of course many more trials, other than 20, and hands other than 20, with more than four players should be carried out in order to safely conclude there is an evident relationship.

3.2 Results and Analysis of Table E

Next, let us define the net amount won, denoted WT, as the total amount won minus the total amount lost. The average amount won, denoted α, is defined as the total amount won, WT, divided by the total amount trials t. Table E gives a summary of the sample data for both WT and α.

TABLE E:Betting Technique

(Players)

WT

(Total Amount Won minus Total Amount Lost)

4 3 2 1

α

(Total Amount Won divided by the total amount of Trials)

4 3 2 1

Exponential

455,917,560

734,812,800

828,274,400

-487,581,800

22,795,878

36,740,640

41,413,720

-24,379,090

Random

87,360 85,650 76,050 52,650 4,368 4,282.5 3,802.5 2,632.5

Hi/Lo 18,100 -18,100 -5,150 9,200 905 -905 -257.50

460

The first and second columns report the total amount won and average amount won applied with the three betting techniques. Again, each column is divided into four separate cells for each case of 1, 2, 3, 4 players vs. the dealer. In column 1, there is a distinct relationship between the total amount won and the number of players sitting at the table under the Random Betting technique. The total amount won by one player, two players, three players, and four players vs. dealer are $52,650, $76,050, $85,650, $87,360 respectively. There is a general positive increase in the total amount won in the number of players at the table increases. This does not mean that the number of player’s at the

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table effect the number of wins or losses on each hand. However, this may show a relationship in the total amount won or lost. The same trend can be seen in column 2 with the average total amount won over each trial as well. Again, there is a positive increase in the average total amount won as number of players at the table increases. However, there is no such trend in exponential and card-counting techniques as in the random betting technique.

3.3 Results and Analysis of Table F

The Overall total amount won and average amount won for each of the three betting techniques can be found in Table F where both WT and α are defined over 4,800 hands.

TABLE F:

Overall Totals Exponential Random Hi/Lo

WT $1,531,422,960 $317,010 $4,050

Α $19,142,787 $3,962.50 $50.63

The overall total amount won for exponential betting is $1,531,422,960, for random betting, $317,010, and for the card-counting technique, $4,050 found. Both amounts are significantly positive and can potentially show they are effective using the zero-memory strategy, but many more hands and trials need to be carried out in order to safely say they are.

3.4 Results and Analysis of Table G

Finally, for t = 1, 2, 3, …, 20, each of the t trials for exponential betting, let h(t) denoted the number of hands it takes for a person ultimate winning to exceed $50,000. Let B(t) be the initial bankroll needed to achieve h(t) and let M(t) be the exact winning amount after t trials. Table G gives a summary of the sample data for the exponential betting technique. Column 1 shows the

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average amount of hands it takes to get to the amount desired. Column 2 reports the average amount of greater than or equal to $50,000 and column 3 reports the average bankroll needed in order to get to the desired amount.

TABLE G:

Cases h(t) M(t) B(t)

4 Players ≈ 12 $377,079 $1,077,775

3 Players ≈ 11 $97,760 $292,470

2 Players ≈ 11 $229,085 $607,760

1 Player ≈ 12 $208,565 $550,780

Overall ≈ 11 $228,112.25 $632,196.25 Column 1 shows that there is a steady trend in the amount of hands it takes to get to the amount greater than $50,000. The average number of hands it takes to get to the desired amount is roughly 11 hands. Most of the cases fall near or on that average. However, looking at columns 2 and 3, there is a different pattern in the amount of bankroll needed to get to the amount desired. For the case four players vs. dealer, there is a general bigger average amount needed and average amount > $50,000 than the cases for 3, 2, and 1 players. In general, there isn’t much of a relationship between the average bankroll needed and average amount desired > $50,000. Overall, Table G didn’t show much of a relationship between the h(t), M(t) and B(t). Using the exponential betting technique, a player’s final bankroll will depend on the amount of hands it takes to get to amount greater than $50K. For example, if it takes one player 17 hands to get to the desired amount and another player 10 hands, the amount that is bet on the 17th hand is $6,553,600 which is much more greater than the amount bet on the 10th hand of $51,200. In all betting techniques that were applied, many more trials and hands need to be conducted in order to prove that any of the relationships are evident

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UML DIAGRAMS FOR THE GAME:1.USECASEDIAGRAM:

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2.CLASSSDIAGRAM:

V3. SEQUENCE DIAGRAM:

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SCREEN SHOTS:

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4 Conclusion

The game of blackjack is one of few casino games where the player’s advantage over the “house” solely depends upon the decisions made by the player for each hand. Basic strategy and zero-memory strategy, defined in section 1.2, are techniques that can be used to aide in a player’s decision. Knowing only the dealer’s up card and the player’s two-card total a player must decide whether to hit, stand, split or double down. In this study, it was found that there is a higher percentage of wins and lower percentage of losses when using the basic strategy or zero-memory strategy compared to no strategy at all. Applying this basic strategy to random, exponential, and card-counting betting techniques, defined in section 2, the random betting technique produced a higher percentage of number of wins than the exponential betting and card-counting technique. From the results and analysis, there is a relationship between the number of players at the table playing and the total amount won a player has using the random betting technique. The exponential betting technique and card-counting technique that were implemented did not relate to the number of players at the table and the total amount won. However, the techniques did yield a positive total amount won that may provide evidence that these techniques are also effective. Many more trials and hands, in addition to the ones carried out in this study, are needed to effectively show that these betting techniques will

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yield a positive final bankroll and give the player that “edge,” over the casino. Using the basic strategy technique applied with the betting strategies may yield a positive total amount overall, and may give the player the advantage over the casino, in order to beat it at it’s own game.

5 Future Research Work

Future work in this area could include changes in the amounts of players at the table. In this case, more players such as 5, 6, even 7 players vs. dealer to see if there is a positive trend in the total amount of money won and the amount a players at the table. Another change could include many more trials and hands. Only 4,800 hands were carried out in this experiment, but if there was 20,000 hands or even 100,000 hands conducted, our analysis of the results will be much more accurate. Also, if the number of hands for each trial was bigger, we can take a look at the results over longer periods of time. Other possible research areas related to this include making the results practical by going to a casino and using the techniques applied in this research, but until I win the lottery, I won’t be doing this anytime soon.

6 Acknowledgments

I would like to thank Prof. Peh Ng, my senior seminar advisor for all of her knowledge, suggestions, and encouragement throughout this project and for making this project possible. I would also like to thank Prof. David Roberts for being my second reader and offering great suggestions in improving my paper. Finally, I would like to thank my family and friends for all their support throughout the process.

7 References

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[1] Baldwin, R., Cantey, W., Maisel, H., and McDermott, J. (1956). The

Optimum Strategy in Blackjack. Journal of the American Statistical

Association, 51 (275), 429-439.

[2] Brisman, A. (1999). Mensa Guide to Casino Gambling: Winning

Ways. New York: Sterling Publishing Co.

[3] Epstein, R. (1977). The Theory of Gambling and Statistical Logic.

New York: Academic Press.

[4] Goodnight, J., and Manson, A. (1975). Optimum Zero-Memory

Strategy and Exact Probabilities for a 4-Deck Blackjack. The

American Statistician, 29 (2), 84-88.

[5] Millman, M. (1983). A Statistical Analysis of Casino Blackjack. The

American Mathematical Monthly, 90 (7), 431-436.

[6] Orkin, M. (1991). Can you Win: The Real Odds for Casino gambling,

Sports Betting, and Lotteries. New York: W. H. Freedman and

Company.

[7] Introduction to the Practice of Statistics. (2006). Statistical Applet.

Simple Random Sample. Retrieved November 28, 2006 from World

Wide Web: http://bcs.whfreeman.com/ips5e/default.asp

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