Tag Archives: Pascal triangle

PASCAL PINBALL MACHINE

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PASCAL PINBALL MACHINE

I call this Pascal Pinball Machine for a good reason. I have seen these pinball machines made by students during school exhibitions several times in the past. Although the decisions are random, the outcomes are orderly. The significance of this post is to illustrate a point that the randomness always operate within a well-defined rule. In this case, the random decisions of the pinball are always defined by the numbers in the Pascal Triangle. You can use this concept to create strategies to win in gambling.

pinball machine

pinball machine

The above picture shows a type of pinball machine that you can build yourself. You will need 10 finishing nails, 5 small cups, a wooden board and a pinball. A pinball can be a marble, ball bearing, golf ball, tennis ball or even table-tennis ball. Generally, I prefer ball bearing. They are heavy, round, smooth and cheap. You can choose the size you want. You can buy the ball bearing from any hardware shop. Hammer the nails half way into the wooden board in the triangular pattern shown, with one nail in the top row, two in the second row, three in the third row and so on, and with enough space for the pinball to go through in between the nails. To make it more user friendly, some pinball machines are enhanced with a spring pedal at the left or right side of the wooden board to spring up the pinball. You can further enhance the pinball machines at the edges and corners with rubber cushions.

To operate the pinball machine, elevate the board at a slight angle about 30 degrees and release the pinball from the top so that it hits the top nail right at the centre. The pinball will be deflected either left or right with equal probability by the first nail. It will then continue falling and hit one of the nails in the second row. In a similar manner, the pinball will be deflected either left or right around the nail.

Because you are building the pinball machine manually, your hand skill is not perfect, so the pinball machine has to be biased in some way. However, you can assume that the pinball machine is fair, unbiased and crafted to perfection for the sake of simplicity.

The result is that the pinball follows a random path, deflecting off one pin in each of the four rows of pins, and ending up in one of the cups at the bottom. The various possible paths are shown in the picture above.

How many random paths are there through your pinball machine, and what are they?

The answer is 16. The mathematical explanation is:

The first row has one pin. So, there are two possible paths for the pinball to go through the first row.

The second row has two pins. Since what happens in the second row is completely independent of what happened in the first row, the number of possible paths the pinball could travel from the top through the second row is (2 x 2) = 4 .

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The fourth row has four pins so the number of possible paths from the top through the fourth row is (2 x 2 x 2 x 2) = 16.

If you drop 16 pinballs into the top of your machine and repeat that event one million times, what is the average number of pinballs per event that will fall into each cup at the bottom? The answer, from left to right, shown in our pinball machine picture above is 1-4-6-4-1. The picture below is known as Pascal Triangle.

pascal triangle

pascal triangle

If you superimpose Pascal Triangle on top of the pinball machine then you can see the corresponding attributes between the two: Each number of Pascal Triangle represents the number of distinct paths that a pinball can take to arrive at that point in the pinball machine.

Pascal Triangle is very useful for analyzing the pinball machine. You have seen the first five rows. Can you see the pattern and guess what the next row of numbers is?

For now, you should be able to appreciate that Pascal Triangle is an orderly description of the outcome of a series of completely random events. Because the pinball has to follow the rules of random decisions, mathematically you can extrapolate the Pascal Triangle to infinity. After you have understood the predictability of pascal triangle in an orderly manner, you can then proceed to test the hypothesis on whether your pinball machine is biased. You do it statistically. You test your sample at 95% confidence interval. You can do the calculation by referring to the statistic tables or you can even do the calculation by using the statistic software.

When you gamble in the casino, always remember that there is orderliness within the random decisions. Random decisions are not random. They have to follow the rules of Pascal Triangle. You can create strategies to take advantage of this orderliness to win in gambling.

KEYPOINTS

1. Rule of Nature is always orderly.
2. Randomness always operates within a well-defined rule.
3. Orderliness and randomness always work hand in hand. Take advantage of them to win in gambling.


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 IN GAMBLING PART 2

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PASCAL TRIANGLE IN GAMBLING PART 2

Unless you master pascal triangle, it is unlikely that you can be a good gambler. You must master pascal triangle if you want to be a good gambler. Pascal triangle gives you the structure to win yet stay away from gambling tilt.

Pascal Triangle is a marvel that develops from a very basic simple formula. Pascal triangle became famous because of many of its patterns.

Before you start looking at patterns, just learn how to write your own pascal triangle. This is for those who do not have flare in mathematics.

Pascal Triangle is formed by starting with an apex of 1. The first row is counted as row zero. Every number below in the triangle is the sum of the two numbers diagonally above it to the left and the right, with positions outside the triangle counting as zero.

pascal triangle

pascal triangle

Now, you may take a look at patterns within the pascal triangle.

PATTERN 1

General patterns found within Pascal Triangle

Heads or Tails, Even or Odd, Black or Red, Big or Small, Banker or Player.

Pascal Triangle can show you how many ways heads and tails can combine. You can then use the pascal triangle to see the odds or probability of any combination.

pascal triangle

For example, if you toss a coin three times, there is only one combination that will give you three heads (HHH), but there are three combinations that will give two heads and one tail (HHT, HTH, THH), also three that give one head and two tails (HTT, THT, TTH) and one for all Tails (TTT). This is the pattern “1,3,3,1” in Pascal Triangle in row 3.

You are assuming that the orders are the same. In other words, (HHT, HTH, THH), (HTH, HHT, THH) and (HTH, THH, HHT) are the same. Bear in mind that in actual gambling they are not the same. You have to make adjustment for that.

Example: What is the probability of getting exactly two heads with 4 coin tosses?

There are 1+4+6+4+1 = 16 (or 2 to the power 4=16) possible results, and 6 of them give exactly two heads. So the probability is 6/16, or 37.5%. Why 37.5%. Why not 50% since two heads out of four. Try to figure it out yourself. (Hint: The rules here is different from the rules in gambling. Here, you win only when the outcome is two heads. You lose when the outcome is one head, three heads and four heads.)

You have seen that Pascal triangle is constructed very simply—each number in the triangle is the sum of the two numbers immediately above it. It is also assumed that you now know how to construct pascal triangle with ease.

Pascal triangle is very useful for finding the probability of events where there are only two possible outcomes. This includes tossing a coin where the outcomes are either head or tail. In mini-dice and Tai-Sai, you have big or small. In roulette, you have black or red, big or small, even or odd. In baccarat, you have banker or player.

For example, if you bet three times in baccarat, there are eight (2x2x2 or 2 to the power 3) possibilities:

BBB BBP BPB PBB PPB PBP BPP PPP

If you look at Row 3 of the triangle, you can see the numbers 1,3,3,1. This tells you that there is only one way of obtaining all BANKERS or all PLAYERS, but three ways of obtaining two BANKERS and one PLAYERS, or two PLAYERS and one BANKER. Translated to probabilities, the chances of the possible outcomes are:

3B—1/8 (one in eight) 2B1P—3/8 2P1B—3/8 3P—1/8 (one in eight)

Refer to Pascal triangle again, and take a look at row 4. Looking at Row 4, you can see that for a set of four bets, one PLAYER and three BANKER is four times as common as having FOUR BANKER and no PLAYER, while a set of four bets with two BANKERS and two PLAYERS are six times as common. There is only one chance in 16 (2 to the power 4) of a set of four having all BANKERS or all PLAYERS. And so on.

COMBINATIONS

The pascal triangle also shows you how many combinations of objects are possible.

Example: You placed 16 bets. How many times would you win only three bets and lost 13 bets? This is a typical gambling scenario.

Answer: go down to row 16 (the top row is 0), and then along 3 places and the value there is your answer, 560.

pascal triangle

PATTERN 2

Patterns found within Diagonals

pascal triangle

The first diagonal is, of course, just “1”s, and the next diagonal has the Counting Numbers (1,2,3, 4,5,6,7,etc).

The third diagonal has the triangular numbers 1,3,6,10,15,21

The fourth diagonal has the tetrahedral numbers 1,4,10,20,35.
The fifth diagonal has the pentagonal numbers.
The sixth diagonal has the hexagonal numbers.

The Fibonacci Series is also found within the diagonals in the Pascal’s Triangle.

The numbers on diagonals of the triangle add to the Fibonacci series, as shown below.

pascal triangle

pascal triangle fibonacci

I will discuss the significance of fibonacci numbers in gambling, nature and life in a separate post.

PATTERN 3

Patterns found within horizontals

pascal triangle horizontal

Notice that each horizontal rows add up to powers of 2 (i.e., 1, 2, 4, 8, 16, etc).

The horizontal rows represent powers of 11 (1, 11, 121, 1331, etc).

Adding any two successive numbers in the diagonal 1-3-6-10-15-21-28… results in a perfect square (1, 4, 9, 16, etc).

PATTERN 4

When the first number to the right of the 1 in any row is a prime number, all numbers in that row are divisible by that prime number. Try it yourself to appreciate.

PATTERN 5

Pattern 5 is combinatoric mathematics. Combinatorics is the science that studies the numbers of different combinations, which are groupings of numbers. Combinatorics is often part of the study of probability and statistics.

Fractal is a term coined by Benoit Mandelbrot in 1975, referring to objects built using recursion, where some aspect of the limiting object is infinite and another is finite, and where at any iteration, some piece of the object is a scaled down version of the previous iteration. A good example of geometric fractal is the Sierpinski Triangle which is an ever repeating pattern of triangles.

sierpinski triangle

PATTERN 6

Pattern 6 is the CATALAN NUMBERS

The Catalan Numbers are a sequence of numbers which show up in many contexts. They were discovered by Leonhard Euler when he was attempting to find a general formula to express the number of ways to divide a polygon with N sides into triangles using non-intersecting diagonals . The Catalan Numbers’ correspondence to the division of polygons is shown below:

pascal catalan

You can see in next Pascal Triangle that each Catalan number is the sum of specific Pascal numbers.(© Dirk Laureyssens, 2004)
I will discuss the significance of catalan numbers in computer science and programming in a separate post.

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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.