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Chapter 1

Chance and Probability

The likelihood that a certain event might take place is impossible to predict, but if the amount

of opportunities is large enough the average outcome for each event occurring can be

predicted. This means that it is possible to predict how many persons within a large city will

get high jacked within the time span of a year, but it is imposable to predict exactly who will

be high jacked

Most things happen by chance which is not the same as saying that it happens by accident;

Erwin Schrdinger concluded that in the overwhelming majority of phenomena where

regularity and invariability have led to the formulation of consistence, was observed within

chance.

Albert Einstein was somewhat uncomfortable with this idea, and said that he cannot believe

that God plays dice with the cosmos. He however found it easier to accept that there were

no laws at all to begin with than to believe Erwin Schrdingers statement.

Chance is derived from French the meaning given of chance are a sequence of events but isnot the same as accident. An accident is a sequence of unpredicted events that cannot be

foreseen, it happens at random without a pattern or meaning. Chance on the other hand

seems to have a pattern and reason of its own. The concept of probability is applicable to

those events that may take place or may not take place according to chance. Biologist Lyall

Watson explains this in his book Beyond Supernature.

I am referring to Lyall Watsons explanation on chance, not just because he is a creative

genius, but also for the reason that he so beautifully explained with a foresight and creative

flair of the working of things. He stated that in order to understand the concept one has to

look at the whole machine which although not made up of the same parts still runs on the

same scientific theories. In order to understand the machines working you have to look at

every component carefully, I believe that this is true and applies to almost every aspect of

existence.

There was however no room for chance or coincidence in the classical world of physics,

within that world everything had a place and direct cause. Isaac Newton believed that it was

only a matter of time before all such causes would be known.

Chance within its context could be something like this, if we should go to a casino and play a

game involving dice, in the hope that we will not lose we take a chance every time we

through the dice.

Two mathematicians Blaise Pascal and Pierre Fermat included chance in this theory, If

something is imposable to happen, let its probability be 0 and if it is certain to happen let it

be 1.With this theory we can make accurate predictions of the outcome of a vast number of falling

dice. We may also define probability as the amount of favourable outcomes to the total

amount of possible outcomes if these remain indifferent, which is pure mathematical, or the

proportion observed within a sample (empirical probability) even the limit of this as size

tends to perpetuity.

Any one of the six faces could turn up, when we threw a perfect regular dice. The

emergence of any one of the six faces after the dice had been thrown is an event. The face

that appears cannot be determined before the event occurs there simply is no physical law

that permits the prediction.

If two is desired in a single though of a dice, it is impossible to know in advance whether the

two will come up, but if this event should take place we can say that the occurrence of theevent is sheer luck. Possible events are those events that might occur and among which at

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least one event must occur, in this case the number of possible events is six. The average

number of successful cases, divided by the number of all possible events gives us the

probability of a chance event occurring. In the case of the dice the lucky cases totals one

and the total of all possible events are six, this yields a probability of 1/6 (= 0.1667) that the

event would probably happen.

A probability must therefore be expressed with a number that lie between 1 (the maximumprobability) and 0 (when there can be no lucky case).

Calculating probability calculates what the chance of an event taking place, 1 the event is

certain to occur and 0 the event is imposable to happen. The formal study of random or

chance events are usually based on independent random variables, the results of which is

applied to empirical data in order to construct statistical tests, that leads to information theory

or game theory.

If we look at combinations and probabilities, the purpose of which is to mathematically

understand the probabilities of events. Logical understanding is derived through the

observation of chance and applying information into statistical format. Combinations are

possibilities, options or arrangements that are open without replication.

How many seating combinations do four people have at a table without repetitions?

Lets assume the table to be around table, which means that any seat taken in between two

people at the edge of this table that these two seating arrangements would be different if

these people would follow each other around the table. If the guests choose their seating

arrangements one at a time, the first guest has four options open to him. Once seated the

second guest has a choice from the remaining three seating positions. Once he has chosen

and taken a seat only two seating positions remain, the third guest chosen his seat, only one

seat now remains and thus the last guest does not have any options, since no options are

available except one.

The possible choices that was open to the first two guests were 4 X 3 and the third guest

had 2 options the last guest had only one option.The total choices were 4 X 3 X 2 X 1 which gives us 24 options. If the guests totalled 6 the

options would have been 6 X 5 X 4 X 3 X 2 X 1 which gives us a figure of 720 arrangements

which might seem to be a large amount. Consider this if the seating arrangement choices

were replaced by 10 people the number of different options would have been much larger a

total of 3, 628, 800 options.

We can see that with the increasing number of people the possible choices increased as

well, we can also see that the number of possible seating arrangements around the table

equals the product of all the whole numbers between one and ten. Lets take the formulation

one step further, by looking at the options open for the arrangements of numbers and letters

on licence plates.

Licence plates should be easy to read, clear from a distance and easy to remember. The

most important fact is there should not be another driver on the road with the same plates

than yours. These set of numbers should only be used or issued once. Lets assume for this

exercise that the licence plate consists of three numbers and three alphabetic characters.

The question we now ask is how many combinations are thus available? In order to clarify

this we have to look at the problem within stages. Arrange the first three letters of the

alphabet in four rows the 26 letters may be used for the first row, this means that there are

26 ways of filling up the position, start with different letters at the beginning of second, third

and fourth rows, this now means that there are 26 ways to fill up the positions in the second

row.

We now know that there are 26 ways in which to write the first and second rows letters,which leaves us with the third row where there are 26 ways of different options of writing the

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third letter on the number plates. This gives us the total number of different combinations of

letters that could be produced on the number plates to be 26 which are 17, 756.

With three letters it will be 10 which gives us a number of 1, 000. However there is a variant

10 gets replaced by zero, the number that can be produces is not higher than 999.

Determined by this law, is the possible number of combinations of digits or letters and can be

calculated when the number of digits or letters are available to us.Mathematicians refer to this group of numbers and symbols as permutations or

arrangements. We have to decide how we have to arrange the 26 objects in threes. The

formula is exact in this case since prediction is not required.

The formula is Dnk=nk

The result of applying the formula is D 26, 3 = 26 = 17, 576.

If you found that fascinating, consider this.

How many options are available when entering the lotto? If we have to choose six possible

numbers from out of a range of numbers 1 to 49, the formula applied will be result in six to

the power of forty nine. The chance of selecting the winning sequence of the lotto is very

slim with the mathematical theory of probability we can ascertain with certainty the outcome

of the possibility of the correct number sequence, in this case to be one out of 13, 841,287,

201.

When we calculate the probability of winning the jackpot in the lotto the chance of winning

would be one over the total amount of tickets sold, this is known as the mathematical hope.

Whoever organised the lottery calculated the mathematical hope before the buyers did.

Establishing how many tickets will be bought and taking in consideration the chance of

selecting the winning sequence as well as the amount aimed to attain as profit after the

winnings has been paid out are the grounds on which the lotto tickets price is set.

That means that should the price be R10, 000 and the total of lotto tickets that was sold are

100 000 then the lotto tickets actual cost is 10c per ticket.

When calculating, R10, 000 1 = 10c.100 000 tickets

After calculating this amount, the organizers will establish how much a ticket will be sold for.

The difference is the profit, and the cost is on the buyers.

Sometimes competition involves those placing a bet to thrust in their knowledge, factors

such as the form of the horse or the prows of a sports team, which will alter the

mathematical hope into their favour. This will result in the value being brought above the

value which would normally apply should it have been solely dependable on chance. In this

case banks are the winners, and have the equation down to a fine art.

More complex concepts are required in several scientific and technical problems. In the

problem of the dice what will be the chance of it turning up a six twice in a row?

The combination chance that this will occur in two throws are one over six for the first throw

and one over six for the second throw, which is one chance out of thirty six possible

combination. The probability of each event taking place is 1/6 for each throw. Therefore the

compound probability is 1/6 X 1/6 which is a result of 1/36.

We can see that the compound probability is much smaller than that of the simple events.

The problems within physics of determining the distribution of molecules in a mixture of

different gasses and being able to forecast which processes will occur spontaneously and

which will take place through an expenditure of energy was solved by the mathematics of

probability.

The mathematical argument can be used to establish the practical impossibility or the

extreme improbability of the molecules of two gasses mixed together spontaneouslyseparating from each other.

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Max Planck based the theory of radiation on the basis of probability and as quantum physics

developed further probability theory extended into atomic theory.

Early scientists believed that all substances were made up of minute particles invisible and

imperceptible to our senses and that each particle considered of a few basic elements.

There is a fascinating property to matter, the transformation pattern is ceaseless. We can

see this in the burning of a log, when it becomes hot air and turn into ash. It can be said nowwith absolute certainty that matter is composed of a number of fundamental elements that

are limited and transformation can occur which depends on the reaction of the smallest

particles into which matter can be subdivided into atoms. This has become common

knowledge today.

Within a substance, atomic properties are the same even if the amount of the property is

large. The concept known as atomic theory first originated two thousand years ago by

Democritus who was a Greek philosopher.

He claimed that, the only existing thing is the atoms and empty space all else is mere

opinion.

His theory was abandoned because at that time it could not be proven and did not lend itself

to the interpretation of phenomena. Towards the end of the 18 th century chemists concluded

that matter indeed must be composed of atoms and that explained a large number of

phenomena. The English Chemist John Dalton in 1808 found Democritus theory useful to

explain the law by which elements unite and produce compounds. At the end of the 19 th

century a well defined theory had developed.

Through studying atomic structure of matter scientists found that to best interpret the

presence of molecules is to define it as small aggregations of atoms which constitute all the

properties of any given substance. Thus Democritus theory was only proven two thousand

years later.

But although the amount of chemical knowledge grew there were still unanswered questions,

it did not lead to the conclusion of weight and size nor was it clear about the relative weightof atoms which was resolved through the study of gasses in the end.

The proton, neutron and electron are the three basic particles in chemistry whose properties

and abilities to form atoms determined by laws of chemistry.

According to principles in physics molecules are a collection of atoms. Chemistry deals with

the isolation of pure substances to establish the structure and interaction between chemicals

that are responsible for new chemicals and new sources of energy. This knowledge of

chemical reactions allows chemists to create new substances through manipulation of

induced circumstances which will lead to an artificially created reaction.

No one has ever seen an atomic nucleus because of the limitation of even the most powerful

microscope but has been proven in theory.

Electrons hold negatively charged electricity, because these electrons orbits the nucleus, we

can now assume that the nucleus holds a positive charge because only a positive charge

can hold to it a negative charge. Thus proving within theory that which we cannot see.

The number of electrons depends on the number of protons and the positively charged

nucleus is balanced by the negatively charged electrons. The electrons determine the

chemical properties and the nucleus constitutes the atomic weight. There is thus a definite

interaction between the proton and electron. This interaction can cause two atoms which

differ by only one proton to be different elements each with a distinct set of properties.

Eleven electrons revolve around a sodium atom and seventeen electrons orbits the chlorine

atom. Numbers is a definite part of existence for the number of electrons determines what it

will be.

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In 1953 the DNA structure was discovered, the master molecule in life. In 1970 Albert V.

Crewe made it possible to see the genetic code along a strand of DNA. We now could see

how life is programmed and this led to the recipe of life. The genetic code of life could now

be examined. It determines the sequence of amino acids in proteins. Most of the details that

involve protein synthesis from these acids are now known, including the mystery of proteins

known as enzymes, which knit together assorted biochemicals into complex machinery ofviruses and bacteria.

With this microscope, the Crewe Microscope, we now could also see a single atom.

I see biology as the science of life. The part which I find most interesting remains the

evolution theories surrounding the origin of the life forms. Lamarck in Darvanism actually

said that alteration in the environment of animals leads to the alteration of needs which in

turn produces modifications within the animals genetic structure and organisation. This

implies that evolution is a responsive process, instead of one that is entirely accidental. The

most significant reference remains the camel. No other animal has the extraordinary features

that this animal has. The camel has astonishing survival features within its genetic blueprint.

Firstly it has very long eye lashes to keep from them the hard desert sand, secondly the

camel has a hump in which is stored essential fats allowing the camel to go without food and

water over long distances. The most unusual feature is the camels legs. The back legs fold

up with the knee to the front, the knee is protected from the hard terrain it occupies by a

cartridge. All camels are born with this feature thus it is included within the blueprint of the

camels genes. Evolution might be more than random events in the view of a desperate

struggle for survival of the species. This theory implies the possibility of purposefulness

within what we understand of evolution.

I agree with Lyall Watson that there is nothing within this to account for developments which

are not only increasing by complexity but are also surprisingly relevant. He believes that

there is a definite force which makes things move and grow in ways that is meaningful and

consequential and that a lot that is happening might well be due to chance but there areinteresting, superimposed patterns on this random world we live on.