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.
Feb 08, 2012 Pascal’s Roulette February 8, 2012 Bengie. Patheos Friendly Atheist. Want more from the Friendly Atheist? Get our FREE Newsletters and special offers! Mar 29, 2019 Pascal was a French mathematician, physicist, inventor, and writer who is credited with inventing the first roulette wheel. It was during Pascal’s younger years that he started to manufacture roulette, which in turn, led him to introduce the mathematical theory of probability.
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.
Now, you may take a look at patterns within the pascal triangle.
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.
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.
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.
Patterns found within Diagonals
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.
I will discuss the significance of fibonacci numbers in gambling, nature and life in a separate post.
Patterns found within horizontals
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).
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 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.
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:
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.
The curve is formed by the locus of a point, attached to a circle (cycle -> cycloid), that rolls along a straight line1). In other words: the combination of a linear (term t) and a circular motion (terms sin t and cos t).
In a Whewell equation the curve can be written as s = sinφ.
The old Greek already knew with this curve.
The value of the parameter 'a' determines the starting point in relation to the circle:
- (ordinary) cycloid
The starting point is situated on the circle (a = 1).
When the starting point is not on the circle, the curve is called a trochoid:
- prolate cycloid (Fr. cycloïde allongée)
The starting point is situated outside the circle (a > 1).
- curtate cycloid (Fr. cycloïde raccourcie)
The starting point is situated inside the circle (a < 1).
When you have a steady hand, you can make your own cycloid on a blackboard, combining a linear and a circular motion.
When a cycloid rolls over a line, the path of the center is an ellipse. For the ordinary cycloid, the result is a circle.
In Holland we use for this curve also the name 'wheel line' 2), being to the track followed by a point on a cycling wheel.
The curve has two remarkable qualities:
The first quality is that the cycloid is the brachistochrone3), that is the curve between two points in a vertical plane, along which a bead needs the shortest time to travel 4). Galilei (who gave the curve its name in 1699) stated in 1638 (falsely) that the brachistochrone has to be the arc of a circle. And in June 1696 Johann Bernoulli challenged his brother Jakob Bernoulli - the both were rivals - to solve the problem. December 1696 Johann 5) repeated his challenge in the 'Acta eruditorum', asking to send solutions before Easter 1697. Besides Johann and Jakob also Leibniz, Newton, and de L'Hôpital solved the problem.
This is one of the first variational problems, to be studied.
As a matter of fact, this curve is the opposite (mirroring in the x-axis) of the shown curve. Without any formula, it can be understood that along this curve the path is faster than along a straight line. For a cycloid the resulting component of the gravity is larger, and so the acceleration and speed, immediately after the start. It is also easy to verify experimentally that the way along a straight line takes more time.
This knowledge can used while skiing: it is faster choosing a way down so that you gain speed, than to avoid the slopes.
The second quality is that the cycloid is the tautochrone (sometimes called: isochrone) 6). This means that a bead along the curve, needs the same time to get down, independent from the starting point. Wonderful! It was Christiaan Huygens who discovered this fact, in 1659. In his treatise 'Horologium Oscilatorium' (1673) he designs a clock with a pendulum with variable length. The pendulum moves between two cheeks, both having the form of a cycloid. Swaying to the outer side, the pendulum shortens. Huygens used the insight that the involute of a cycloid is the same cycloid (of course also valid for the evolute).
By this construction the irregularity in a normal pendulum was compensated. For a normal pendulum then, the time of oscillation is only in a first approximation independent of the position aside 7). Huygens was quite impressed, he wrote above the prove: 'magna nec ingenijs investigata priorum', to be translated as: 'this is something great, and never before investigated by a genius'.
In his experiments Huygens used also the involute of a circle in his pendulum clock to approximate the cycloid path.
However, use of the tautochrone principle in designing pendulum clocks posed too many mechanical problems, to make it common.
The tautochrone quality can also be read in Moby Dick, the book from Herman Melville, in a treatise about a kettle to boil whale oil.
Some interesting properties of the cycloid are the following:
- its radial is the circle
the isoptic of the cycloid is a trochoid
Cusa was the first to study the curve in modern times, when trying to find the circle's area. Mersenne (1599) gave the first proper definition of the cycloid, he tried to find the area under the curve but failed. He posed the question to Roberval, who solved it in 1634. Later, Torricelli found the curve's area, independently.
Descartes found how to draw a tangent to the cycloid, he challenged Roberval to find the solution, Roberval failed, but Fermat succeeded. Also Viviani found the tangent.
In August 1658 Pascal published a challenge, offering two prizes, under the name of Amos Dettonville. He posed 9 questions on the cycloid, asking for the area and the center of gravity of its segment.
It is said that for Pascal studying the curve was a good diversion from his severe toothache.
Wallis and Lalouére entered, both were not successful. Sluze, Ricci, Huygens, Wren and Fermat did not enter the competition, but all wrote their solution to Pascal. On October 10 1658, Pascal published his own solutions, together with an extension of Wren's result.
The name of the paper was: 'Histoire de la Roulette, appelé autrement Trochoide ou Cycloide'.
Desargues proposed teeth for gear wheels, in the form of a cycloid (about 1635).
Jakob and Johann Bernoulli showed (1692) that the cycloid is the catacaustic of a circle, where light rays come from the circumference.
A cycloid arch, with rays perpendicular to the x-axis, results in two cycloid arches.
So the cycloid was very popular among 17th century mathematicians. That's why later the curve has been given the names of quarrel curve, Helen of Geometers, and apple of discord9).
In the Piano Museum in Hopkinton10) one finds a piano, whose back edge has the form of a cycloid. The maker, Henry Lindeman, named the instrument 'the Cycloid Grand', in the late 1800s.
But seen from the upside you see that its form differs from a real cycloid:
Gear wheels have a cycloid form, they can be approximated by a series of circular arcs. Also numeric tables can be used, as George Grant's Odontograph, which is also the name for an instrument for laying off the outlines of the teeth of the gear wheels.
The curve is a point-roulette.
Now the point being followed is not lying on the circle. When the point lays outside the circle, the curve is called a prolate cycloid (orextended cycloid). When the point lays inside the rolling circle, the curve is called a curtate cycloid (or contracted cycloid). The latter curve is followed by the valve of a bike. That's where the name valve curve for the cycloid is from.
The first to study the curve were Dürer (1525) and Römer (1674).
With a Voith-Schneider Propeller (VSP), first tested in 1927, a ship is able to maneuver precisely, and to move aside. The propellers rotate around an axis, vertical to the motion, so that they follow the path of a cycloid. The position of the blades determine the direction of the vessel, and it is based on the same principle of a fish's fin action.
The propeller is called a cycloidal or trochoidal propeller.
1) Let there be a circle with center (0,R) and a point (p, 0) as starting point to roll. Then the coordinates of the cycloid, as function of the rolled angle t are
2) In Dutch: radlijn
3) Brakhisto (Gr.) or brachus (Lat.) = short, chronos (Gr.) = time
4) At height y the bead becomes a velocity √gy, so that minimizing the travel time means minimizing the integral
Solving this equation leads via differential equation y (1 + y'2) = c to the cycloid.
5) In that period professor in mathematics in Groningen, Holland Philipp gruissem.
6) Tauto = equal, chronos = time: the curve to be followed in equal time.
7) The correct relation is given by a complete elliptical integral of the first kind.
8) In English: A history on the roulette, also named trochoid or cycloid.
9) Helen and the apple of discord refer to the Trojan war.
Dutch for quarrel curve: kibbelkromme.
Pascal Haas Roulette
10) Hopkinton, Mass., about 1/2 hr. west of Boston, see The Piano Museum website.
Blaise Pascal Roulette Machine
11) Trochus (Lat.) = hoop.
Sometimes the meaning of cycloid and trochoid is interchanged: trochoid for the general case, cycloid only for the situation that the starting point is lying on the circle.