PASCAL TRIANGLE IN GAMBLING PART 2

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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