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2. The most famous one is the Martingale strategy, which was created by an 18th-century casino owner. It is a fairly straightforward system and can be easily applied to online casinos. The basic idea is to double your stake each time you lose a bet, so when you eventually win, you will recoup all previous losses.

Hobby softwares ( lotteries - loto, keno, euromillions - soccer and football) and business softwares ( labelling and barcoding ). Roulette Martingale Strategy; Simple Gun and Run Martingale Simple Paroli System; Roulette and craps are probably the two casino games with the longest pedigree. We know that dice were first used as a serious incantation to find out what the gods had in store for people.

STATE UNIVERSITY COLLEGE AT BUFFALO

Department of Mathematics

Course Revision

I. Number and Title of Course

MAT 107 Casino Gambling

II. Reasons for Revision

We have updated the bibliography to better address and reflect the current work in this area, and we have redefined the major objectives of the course in terms of student outcomes. The course continues to serve the following purposes in our program:

A. To provide an intensive encounter with the behavior of the phenomenon of chance that pervades everyone’s life and that is an essential characteristic of the world in which we live. Life is said to be a school of probability.

B. To develop an intelligent approach toward the phenomenon of chance and an understanding of the calculation of risks involved in making a decision. Casino gambling games offer an excellent training opportunity for such knowledge just as Geometry is an excellent training opportunity for learning logic. Indeed, in all the affairs of life we must make decisions that are gambles because risk is involved.

C. To present the fundamental elements of Combinatorial Analysis and of Probability Theory, which is perhaps the single most important model of reality, in order to produce well-educated citizens who know and understand the basic facts of chance and the way it works.

D. To critically examine casino gambling, a topic that is currently gaining popularity and that, besides being a source of entertainment, offers a splendid model for a study of the workings of probability theory, leading to a comprehension of chance that is of real value when applied to many other phases of life.

E. To eliminate many common and widely held superstitions about the phenomenon of chance and the laws that govern it.

III. Major Objectives of the Course

A. Students will examine the pervasive phenomenon of chance and the laws of probability according to which this phenomenon behaves.

B. Students will develop the abilities of logical analysis and critical thinking concerning possibilities and decisions open to them by means of a thorough examination of the rules governing popular games of chance in gambling casinos.

C. Students will demonstrate how the laws of probability and mathematical expectation, together with careful analysis, can uncover the precise value of the player’s disadvantage in various gambling situations.

D. Students will understand the power and inexorability of the advantage held by gambling operators through their favorable House Percentages on each bet made.

E. Stduents will experience the contrasting emotional and intellectual reactions held by one who faces a real situation governed by chance and in which s/he must make a decision.

IV. Topical Outline

A. Nature of the phenomenon of chance and the laws of probability

1. Objective view of chance, and its measure, probability

2. Subjective view of chance, and its measure, betting odds

3. Law of Large Numbers (Law of Averages) and fallacies concerning it:

a. Doctrine of the maturity of chances

b. Doctrine of runs of luck

c. Fallacy of the small sample.

4. Permutations and combinations

5. Probability laws of addition and multiplication

6. Mathematical expectation

B. Roulette 1. Layout, rules of play and betting

2. Computation of House Percentages

3. Betting systems as attempts to control chance:

a. Martingale

b. Great Martingale

c. D’Alembert system (Up-Down system)

d. Cancellation system

C. Craps 1. Layout, rules of play and betting

2. Computation of House Percentages

3. Comparison of advantages and disadvantages of various available bets

D. Blackjack (Twenty-One) 1. Layout, rules of play and betting

2. Basic Strategy to reduce the casino advantage

3. Introduction to variable strategies

E. Other casino games (optional) 1. Baccarat

2. Wheel of Fortune

3. Keno

4. Slot Machines

F. Applications of the probability model’s principles to situations other than casino gaming

V. Bibliography

Allen, J. Edward. The Basics of Winning Blackjack. New York: Cardoza Publishing, 1992.

Allen, J. Edward. Winning Craps for the Serious Player. New York: Cardoza Publishing, 1993.

Barnhart, Russell T. Beating the Wheel. Secaucus, N. J.: Lyle Stuart (Carol division), 1992.

Chambliss, Carlson R. and Roginski, Thomas C. Fundamental of Blackjack. Las Vegas, Nevada: GBC Press, 1990.

Epstein, Richard A. The Theory of Gambling and Statistical Logic (2nd edition). New York: Academic Press, 1995.

Freund, John E. Introduction to Probability. Mineola, N. Y.: Dover, 1993.

Gollehon, John. All About Roulette. New York: Perigee Books, 1988.

Griffin, Peter A. The Theory of Blackjack (5th edition). Las Vegas, Nevada: Huntington Press, 1996.

Humble, Ph.D., Lance, and Cooper, Ph.D., Carl. The World’s Greatest BlackjackBook (revised edition). New York: Doubleday, 1987.

Hutchinson, Robert J. The Absolut Beginner’s Guide t Gambling. New York: Pocket Books, 1996.

Levinson, Horace C. Chance, Luck and Statistics. New York: Dover, 1963.

Patrick, John. John Patrick’s Craps. Secaucus, N. J.: Lyle Stuart (Carol division), 1991.

Reber, Arthur S. The New Gambler’s Bible. New York: Random House (Crown division), 1996.

Revere, Lawrence. Playing Blackjac as a Business (new revised edition). Secaucus, N. J.: Lyle Stuart (Carol division), 1980.

Ross, Sheldon M. A First Course in Probability (4th edition). Englewood Cliffs, N. J.: Prentice Hall, 1994.

Scarne, John. Scarne’s New Complete Guide to Gambling. New York: Simonand Schuster, 1974.

Scoblete, Frank. Spin Roulette Gold. Chicago: Bonus Books, 1997.

Silberstang, Edwin. The Winner’s Guide to Casino Gambling (3rd edition). New York: Penguin Books USA (Plume division), 1997.

Sklansky, David. Getting the Best of It (revised edition). Henderson, Nevada: Two Plus Two Publishing, 1989.

Thomason, Walter. The Ultimate Blackjack Book. Secaucus, N. J.: Lyle Stuart

(Carol division), 1997.

Thorp, Edward O. Beat the Dealer (revised edition). New York: Random House (Vintage Books division), 1966.

Thorp, Edward O. The Mathematics of Gambling. Secaucus, N. J.: Lyle Stuart, 1984.

Weaver, Warren. Lady Luck, the Theory of Probability. Garden City, N. Y.: Doubleday, 1963.

Wilson, Allan N. The Casino Gambler’s Guide (enlarged edition). New York: Harper & Row, 1970.

Wong, Stanford. Professional Blackjack. La Jolla, Calif.: Pi Yee Press, 1994.

VI. Presentation and Evaluation

A. Lectures, demonstrations and discussions will be used. An essential element in the classroom will be the actual playing of the major casino games so that the student gains first-hand encounters with the phenomenon of chance and the problems of making a decision in the face of uncertainty. This will be done initially to acquaint the student with the rules of play and betting procedures, and it will also be done after a critical examination of the game is made using the theory of probability so that the student may actually see this knowledge put to real use in the gambling decisions he or she then makes.

B. Evaluation will be made through written examinations of the student’s knowledge of the casino games and the elements of probability theory, and of how the latter affects decisions made in the former.

VII. Prerequisite

3 years of Regents high school mathematics or equivalent.

VIII. Credit

3 credits: (3:0)

A martingale is any of a class of betting strategies that originated from and were popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads and loses if it comes up tails. The strategy had the gambler double the bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake.

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Since a gambler will almost surely eventually flip heads, the martingale betting strategy is certain to make money for the gambler provided they have infinite wealth and there is no limit on money earned in a single bet. However, no gambler possess infinite wealth, and the exponential growth of the bets can bankrupt unlucky gamblers who chose to use the martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins a small net reward, thus appearing to have a sound strategy, the gambler's expected value remains zero because the small probability that the gambler will suffer a catastrophic loss exactly balances with the expected gain. In a casino, the expected value is negative, due to the house's edge. Additionally, as the likelihood of a string of consecutive losses occurs more often than common intuition suggests, martingale strategies can bankrupt a gambler quickly.

The martingale strategy has also been applied to roulette, as the probability of hitting either red or black is close to 50%.

Intuitive analysis[edit]

The fundamental reason why all martingale-type betting systems fail is that no amount of information about the results of past bets can be used to predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption that the win-loss outcomes of each bet are independent and identically distributed random variables, an assumption which is valid in many realistic situations. It follows from this assumption that the expected value of a series of bets is equal to the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet times the probability that the player will make that bet. In most casino games, the expected value of any individual bet is negative, so the sum of many negative numbers will also always be negative.

The martingale strategy fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets (which is also true in practice).^[1] It is only with unbounded wealth, bets and time that it could be argued that the martingale becomes a winning strategy.

Mathematical analysis[edit]

The impossibility of winning over the long run, given a limit of the size of bets or a limit in the size of one's bankroll or line of credit, is proven by the optional stopping theorem.^[1]

However, without these limits, the martingale betting strategy is certain to make money for the gambler because the chance of at least one coin flip coming up heads approaches one as the number of coin flips approaches infinity.

Mathematical analysis of a single round[edit]

Let one round be defined as a sequence of consecutive losses followed by either a win, or bankruptcy of the gambler. After a win, the gambler 'resets' and is considered to have started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence of independent rounds. Following is an analysis of the expected value of one round.

Let q be the probability of losing (e.g. for American double-zero roulette, it is 20/38 for a bet on black or red). Let B be the amount of the initial bet. Let n be the finite number of bets the gambler can afford to lose.

The probability that the gambler will lose all n bets is qⁿ. When all bets lose, the total loss is

${\displaystyle \sum _{i=1}^{n}B\cdot 2^{i-1}=B(2^n-1)}$

The probability the gambler does not lose all n bets is 1 − qⁿ. In all other cases, the gambler wins the initial bet (B.) Thus, the expected profit per round is

${\displaystyle (1-q^n)\cdot B-q^n\cdot B(2^n-1)=B(1-(2q{)}^n)}$

Whenever q > 1/2, the expression 1 − (2q)ⁿ < 0 for all n > 0. Thus, for all games where a gambler is more likely to lose than to win any given bet, that gambler is expected to lose money, on average, each round. Increasing the size of wager for each round per the martingale system only serves to increase the average loss.

Suppose a gambler has a 63 unit gambling bankroll. The gambler might bet 1 unit on the first spin. On each loss, the bet is doubled. Thus, taking k as the number of preceding consecutive losses, the player will always bet 2^k units.

With a win on any given spin, the gambler will net 1 unit over the total amount wagered to that point. Once this win is achieved, the gambler restarts the system with a 1 unit bet.

With losses on all of the first six spins, the gambler loses a total of 63 units. This exhausts the bankroll and the martingale cannot be continued.

In this example, the probability of losing the entire bankroll and being unable to continue the martingale is equal to the probability of 6 consecutive losses: (10/19)⁶ = 2.1256%. The probability of winning is equal to 1 minus the probability of losing 6 times: 1 − (10/19)⁶ = 97.8744%.

The expected amount won is (1 × 0.978744) = 0.978744.
The expected amount lost is (63 × 0.021256)= 1.339118.
Thus, the total expected value for each application of the betting system is (0.978744 − 1.339118) = −0.360374 .

In a unique circumstance, this strategy can make sense. Suppose the gambler possesses exactly 63 units but desperately needs a total of 64. Assuming q > 1/2 (it is a real casino) and he may only place bets at even odds, his best strategy is bold play: at each spin, he should bet the smallest amount such that if he wins he reaches his target immediately, and if he doesn't have enough for this, he should simply bet everything. Eventually he either goes bust or reaches his target. This strategy gives him a probability of 97.8744% of achieving the goal of winning one unit vs. a 2.1256% chance of losing all 63 units, and that is the best probability possible in this circumstance.^[2] However, bold play is not always the optimal strategy for having the biggest possible chance to increase an initial capital to some desired higher amount. If the gambler can bet arbitrarily small amounts at arbitrarily long odds (but still with the same expected loss of 10/19 of the stake at each bet), and can only place one bet at each spin, then there are strategies with above 98% chance of attaining his goal, and these use very timid play unless the gambler is close to losing all his capital, in which case he does switch to extremely bold play.^[3]

Alternative mathematical analysis[edit]

The previous analysis calculates expected value, but we can ask another question: what is the chance that one can play a casino game using the martingale strategy, and avoid the losing streak long enough to double one's bankroll.

As before, this depends on the likelihood of losing 6 roulette spins in a row assuming we are betting red/black or even/odd. Many gamblers believe that the chances of losing 6 in a row are remote, and that with a patient adherence to the strategy they will slowly increase their bankroll.

In reality, the odds of a streak of 6 losses in a row are much higher than many people intuitively believe. Psychological studies have shown that since people know that the odds of losing 6 times in a row out of 6 plays are low, they incorrectly assume that in a longer string of plays the odds are also very low. When people are asked to invent data representing 200 coin tosses, they often do not add streaks of more than 5 because they believe that these streaks are very unlikely.^[4] This intuitive belief is sometimes referred to as the representativeness heuristic.

Anti-martingale[edit]

In a classic martingale betting style, gamblers increase bets after each loss in hopes that an eventual win will recover all previous losses. The anti-martingale approach, also known as the reverse martingale, instead increases bets after wins, while reducing them after a loss. The perception is that the gambler will benefit from a winning streak or a 'hot hand', while reducing losses while 'cold' or otherwise having a losing streak. As the single bets are independent from each other (and from the gambler's expectations), the concept of winning 'streaks' is merely an example of gambler's fallacy, and the anti-martingale strategy fails to make any money. If on the other hand, real-life stock returns are serially correlated (for instance due to economic cycles and delayed reaction to news of larger market participants), 'streaks' of wins or losses do happen more often and are longer than those under a purely random process, the anti-martingale strategy could theoretically apply and can be used in trading systems (as trend-following or 'doubling up'). (But see also dollar cost averaging.)

Martingale Chart

See also[edit]

References[edit]

Martingale Chain

1. ^ ^abMichael Mitzenmacher; Eli Upfal (2005), Probability and computing: randomized algorithms and probabilistic analysis, Cambridge University Press, p. 298, ISBN978-0-521-83540-4, archived from the original on October 13, 2015
2. ^Lester E. Dubins; Leonard J. Savage (1965), How to gamble if you must: inequalities for stochastic processes, McGraw Hill
3. ^Larry Shepp (2006), Bold play and the optimal policy for Vardi's casino, pp 150–156 in: Random Walk, Sequential Analysis and Related Topics, World Scientific
4. ^Martin, Frank A. (February 2009). 'What were the Odds of Having Such a Terrible Streak at the Casino?'(PDF). WizardOfOdds.com. Retrieved 31 March 2012.

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