Gambling, Dice Theory, Blaise Pascal and Chevalier De Mere

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.


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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