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Gambling, Dice Theory, Blaise Pascal and Chevalier De Mere

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Gambling, Dice Theory, Blaise Pascal and Chevalier De Mere

Archeologists have discovered dice several thousand years ago. However, modern dice games only grew popular in the middle ages. Traditionally, the chances of winning in gambling were very much evaluated with a combination of wishful thinking as well as some sort of logical reasoning. The logical reasoning may be correct or it may be flawed. The logical reasoning may be derived from empirical observation or mathematics or even both.

dice theory

Historically, the experience of Chevalier de Mere is a landmark case to illustrate this logical reasoning in the dice theory.

Chevalier de Mere was a mid-seventeenth century high-living nobleman and gambler who attempted to make money gambling with dice. Probability theory had not yet been developed during that period, but Chevalier de Mere made money by betting that he could roll at least one 6 on four rolls of one die. Empirical experience led him to believe that he would win more times than he would lose with this bet. In other words, if a six appear within the first four rolls of one die, Chevalier de Mere win the bet. If no six appear within the first four rolls, he loses the bet. Today we know that the probability of winning this bet is 1 – (5/6)4, or 51.8%.

dice theory

When folks would no longer bet on this proposition with Chevalier de Mere, he created a new proposition. He began to bet he would get a total of 12 (or a double 6) on twenty-four rolls of two dice. This seemed like a good bet, but he began losing money on it. That year was around 1654. Chevalier de Mere suffered severe financial losses for assessing incorrectly his chances of winning in this proposition of the dice game. Contrary to the ordinary gambler, he pursued the cause of his failure with the help of Blaise Pascal. Together with Pierre de Fermat, a fellow mathematician of Blaise Pascal, the trio became famous because in the process they had sown the seeds in the development of the modern theory of probability. Let us take a look at what happened.

First proposition: Roll a single die 4 times and bet on getting a six.

Remember the dice has only six numbers. Not two or seven or ten. The base of a die is six.

Rolling a single die once leads to precisely one of 6 possible outcomes: Exactly one of the numbers 1,2,3,4,5,6 will be rolled. The die is described as a fair die if each of these outcomes is equally likely. A fair die has equal outcome. On the contrary, a distorted die has unequal outcome. A distorted die is bias. Players of dice games usually assume that the dice they are using are fair. So let us assume this too.

If you roll a die 4 times, then the total number of all possible outcomes is

6 x 6 x 6 x 6 = 1296

Out of these there are

5 x 5 x 5 x 5 = 625

outcomes with not a single 6 in them.

Thus, if you bet on getting at least one 6 when rolling a die 4 times, there are

625 possibilities of losing, and
1296 – 625 = 671 possibilities of winning

This means that your chances of winning with this game are higher than our chances of losing. So, Chevalier de Mere is correct in evaluating his chances of winning.

(Note: Let’s take a look at another argument. The chance of getting a 6 in a single throw is 1 out of 6. Therefore, the chance of getting a 6 in 4 rolls is 4 times 1 out of 6. That is 2 out of 3. This mathematical reasoning is wrong. Can you figure it out yourself why it is wrong. Hint: Read Chevalier de Mere’s proposition again. He said he could roll one six on four roll of one die.)

dice theory

Second proposition: Roll two dice 24 times and bet on getting a double six.
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He assessed his chances of winning as follows. The chance of getting a double six in one roll is 1 out of 36. Therefore, the chances of getting a double six in 24 rolls is 24 times 1 out of 36; i.e. 2 out of 3.

To his painful surprise the Chevalier ended up loosing badly with the second gamble. He was desparate for an explanation, and so he sought help from one of the great thinkers of his time, Blaise Pascal (1623-1662). After a careful analysis, Pascal was able to point out Chevalier’s error.

Rolling two dice once leads to one of 36 possible outcomes, namely all possible outcomes of rolling die number 1 combined with all possible outcomes of rolling die number two. Thus, if you roll two dice 24 times, then the total number of possible outcomes is

36 x 36 x … x 36 (36 to the power 24)

which is approximately 22,452,257,707,350,000,000,000,000,000,000,000,000.

Out of these there are

35 x 35 x … x 35 (35 to the power 24)

which is approximately 11,419,131,242,070,000,000,000,000,000,000,000,000 outcomes with no double 6.

Thus, if you gamble on getting at least one double 6 when rolling two dice 24 times, there are approximately

11,419,131,242,070,000,000,000,000,000,000,000,000 possibilities of losing, and
22,452,257,707,350,000,000,000,000,000,000,000,000
minus
11,419,131,242,070,000,000,000,000,000,000,000,000 = 11,033,126,465,280,000,000,000,000,000,000,000,000
possibilities of winning

Pascal figured that the probability of not rolling a total of 12 in twenty-four rolls is (35/36)24, or about 50.9%. Hence, in the long run, this would be a losing proposition for Chevalier de Mere.

This means that the chances of winning with this proposition are lower than the chances of losing, as the Chevalier De Mere learnt the dice theory the hard way.

dice theory

Pascal got interested in analyzing other gambling games, and got Pierre de Fermat to work with him. In the process he discovered a fundamental principle for assessing the probability for a certain event, amongst a collection of possible events, to occur. This fundamental principle is just as valid now as it was then. It is broadly used and constitutes a landmark point in the development of the theory of probability. It can be said that the formal study of probability was launched by two mathematicians and a gambler. Not surprisingly, Pascal’s Triangle is a useful tool in probability theory.

Finally, please be aware that logical reasoning in gambling is very often flawed. In professional gambling, you need to have a flare in mathematics to do well.

KEYPOINTS

1. The keypoint to learn here is that logical reasoning very often is flawed.

2. In professional gambling, the basic requirement is a flare in mathematics.


Blaise Pascal Stories

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Blaise Pascal Stories

We all come to know of Blaise Pascal because of Pascal Triangle. We all also know that pascal triangle is used to determine the probability of certain outcomes.

Hydraulic Press and Syringe

After discovering the significance of the various patterns in pascal triangle, Blaise Pascal continued his work and made major contributions in science and mathematics. Among his achievements, Blaise Pascal experimented with atmospheric pressure and showed that atmospheric pressure as measured by the mercury in the barometer decreases as altitude increases, and also changes as the atmospheric temperature changes. Pascal made a valuable contribution in developing both hydrostatics and hydrodynamics. He showed that the pressure within a confined liquid is transmitted uniformly through the liquid in all directions, regardless of the area to which the pressure is applied. This is known as Pascal’s Law and is the principle behind the hydraulic press. During these experiments with fluids, Pascal designed the hydraulic press and invented the syringe.

The Cycloid

Pascal also studied another interesting phenomenon known as cycloid. Cycloid refers to the curve formed by a point on the circumference of a circle as the circle rolls along a straight line. Pascal’s discovery of several physical and mathematical properties of the cycloid was an important step towards the subsequent development of calculus by his fellow mathematicians.

Theory of Probability

Pascal also collaborate with another mathematician, Fermat, on the Theory of Probability. Regular correspondence between Pascal and Fermat by mail showed that both of them contributed equally in the development of the probability theory. Although their investigations were mainly done on various gambling situations, this theory of probability has a significant number of applications. It is the basis of all insurance schemes and it is of great value to many other branches of science such as quantum physics, where the behaviour of particles can be described using the theory of probability.

Gambling

Pascal attended parties where gambling was being organized, and probably became attracted to this lifestyle. Then one day in 1654, Pascal had a narrow escape from death, when the horses pulling his carriage bolted. The horses died in the accident, but Pascal escaped unhurt. Pascal became religious after this incident.

Religion

From the age of thirty-one until the day of his death, at the age of thirty-nine, much of Pascal’s work was devoted to religious writings. He wrote a famous series of 18 letters known as the ‘Provincial Letters,’ considered by critics to mark the beginning of modern French prose. Pascal also wrote the outstanding book Pensees (French for ‘thoughts’) in which he argues the case for his religious beliefs. In his Pensees, Blaise Pascal recognized that man could not acquire all the knowledge by his own wisdom. Pascal even went to the extent of coining the term “Pascal’s Wager ” in which he reviewed his thinking in terms of theory of probability to the uncertainties in life.

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Pascal Triangle and Gambling

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Pascal Triangle and Gambling

PREFACE

Pascal triangle can help you win in gambling. You will win 100% of your series because If you understand pascal triangle enough, you will find that patterns are predictable. You can use these predictable patterns in pascal triangle to win 100% of your series. (Do not confuse these predictable patterns from the unpredictable score-charts provided by the casino)

Capitalize on the predictable patterns provided by the pascal triangle and win 100% of your series.

PASCAL TRIANGLE AND ME

As a kid I learned pascal triangle from my elder brother. As a schoolboy I used to tease my teachers with pascal triangle.

And, in a school exhibition I led a team of four fellow students to do a pin-ball exhibition where the outcome can be estimated by pascal triangle. The public visits the exhibition and throw the pinball. The outcome is recorded in the blackboard. Here, the public see how the outcomes approach the values predicted by the pascal triangle. Our team will answer the questions from the public. Overall, I can see that the public is quite impressed with this project. I will discuss more on this project in my subsequent post.

INTRODUCTION TO PASCAL TRIANGLE

In a nutshell, this is pascal triangle.

pascal triangle

Pascal triangle is named after the French mathematician and philosopher Blaise Pascal (1623-62), who wrote a Treatise on the Arithmetical Triangle describing the properties of this peculiar triangle now known as pascal triangle. However, Blaise Pascal did not discover the sequence of numbers that bears his name. The origin is believed to be hundreds of years earlier in various part of the world. More importantly, Blaise Pascal make popular the sequence in the 17th century from his research and help his French nobleman in improving the betting odds .

Historically, 10th century Indian mathematicians described this array of numbers as useful for representing the number of combinations of short and long sounds in poetic meters. During the eleventh century in Persia, the pascal triangle also appears in the writings of Omar Khayyam. He was an astronomer, poet, philosopher, and mathematician. The Chinese mathematician Chu Shih Chieh depicted the triangle and indicated its use in providing coefficients for the binomial expansion in his 1303 treatise “The Precious Mirror of the Four Elements”. Below is a reproduction of the triangle from Chu Shih Chieh, in Chinese numerals.

chinese pascal triangle
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Although ancient Chinese first developed the triangle, Blaise Pascal was the first person to discover the significance of the patterns it contained.

Blaise Pascal was born on 19 June 1623, in the rural town of Clermont-Ferrand in France. At the age of three his mother passed away. Subsequently, the family shifted to Paris. Blaise Pascal was generally not in good health over the major part of his life span. His father, a tax collector was worried that learning mathematics might worsen his health. However, Blaise Pascal was blessed with a talent in mathematics.

At the age of 14, Blaise Pascal started attending weekly lectures in mathematics. It was from these weekly meetings of mathematicians that the French Academy of Sciences later formed. At the age of 16, Blaise Pascal wrote a paper on conic sections which was acknowledged by his fellow mathematicians as the most powerful and valuable contribution that had been made to the discipline of mathematics since the days of Archimedes. This paper laid the foundation for the modern treatment of conic sections.

Historical records showed that Blaise Pascal’s work on the triangle originated from the popularity of gambling. A French nobleman had approached Blaise Pascal with a problem about gambling with dice. Pascal became interested in the philosophical problem of how to make decisions involving uncertain events. His studies lead to his writing of Traite du Triangle Arethmetique which was the first book on probability theory. The French version of Traite du Triangle Arethmetique when translated into English is known as Pascal’s Arithmetical Triangle. There was evidence that Pascal shared this problem with another famous mathematician known as Fermat.

Pascal made several other important contributions to the history of mathematics, including the first digital calculator, which he designed to help his father in collecting tax. The addition of French currency was difficult, because the currency consisted of livres, sols, and deniers, with 12 deniers in a sol and 20 sols in a livre. Pascal’s machine, known as the Pascaline, was never a success. As many as fifty types and variations were produced, but the machine did not sell well.

Blaise Pascal passed away at the age of 39. After the death of Blaise Pascal, mathematicians have found numerous patterns in Pascal triangle. Some of the most interesting patterns are obtained by coloring in multiples of various numbers in Pascal triangle. The results form endlessly repeating patterns called geometric fractals. Geometric fractals is significant because they are predictable patterns. These predictable patterns can be exploited in gambling.

PASCAL TRIANGLE, PROBABILITY AND EVEN-ODD GAMBLING

Pascal triangle is a very interesting even-odd phenomenon. As you have seen, it takes a few hundred years for mathematicians to solve the riddle of pascal triangle. How long do you think it would take gamblers to solve the riddle of even-odd gambling?

As long as gambling activities remain attractive and popular, gamblers would try to find a system that will allow them to gain the edge. Some of the greatest minds in history have tried to devise a system for beating the casino games. You have seen that the late 17th century French mathematician Blaise Pascal was asked by a friend for help with predictable patterns in wagering propositions.

It has been said that Albert Einstein also studied the problem of how to beat the game of Roulette. After researching the problem, Albert Einstein concluded that it could not be done and he was quoted as saying, “The only way to beat Roulette is to steal the money when the dealer is not watching.” In a sense, he was correct. His point was that there is no way to apply a mathematical configuration of bets to overcome the house edge.

The saying of all casino owners throughout the world is:

“All gamblers will lose to me because it is OUR game”

In other words, it is the game of the casino owners, not the gamblers.