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Introduction

Probability theory is a branch of mathematics concerned with

determining the likelihood that a given event will occur. This

likelihood is determined by dividing the number of selected events

by the number of total events possible. For example, consider a

single die (one of a pair of dice) with six faces. Each face contains a

different number of dots: 1, 2, 3, 4, 5, or 6. If you role the die in a

completely random way, the probability of getting any one of the

six faces (1, 2, 3, 4, 5, or 6) is one out of six.

Probability theory originally grew out of problems encountered by

seventeenth-century gamblers. It has since developed into one of

the most respected and useful branches of mathematics with

applications in many different industries. Perhaps what makes

probability theory most valuable is that it can be used to determine

the expected outcome in any situation—from the chances that a

plane will crash to the probability that a person will win the lottery.



Part 1

Part 1 (a):

History Of Probability Theory

Probability theory was originally inspired by gambling problems. The earliest work on the

subject was performed by Italian mathematician and physicist Girolamo Cardano (1501–

1576). In his manual Liber de Ludo Aleae, Cardano discusses many of the basic concepts

of probability complete with a systematic analysis of gambling problems. Unfortunately,

Cardano's work had little effect on the development of probability because his manual did

not appear in print until 1663—and even then received little attention.

In 1654, another gambler named Chevalier de Méré invented a system for gambling that

he was convinced would make money. He decided to bet even money that he could roll at

least one twelve in 24 rolls of two dice. However, when the Chevalier began losing money,

he asked his mathematician friend Blaise Pascal (1623–1662) to analyze his gambling

system. Pascal discovered that the Chevalier's system would lose about 51 percent of the

time.

Pascal became so interested in probability that he began studying more problems in this

field. He discussed them with another famous mathematician, Pierre de Fermat (1601–

1665) and, together they laid the foundation of probability theory.

The probability of rolling snake eyes (two ones) with a pair of dice is 1 in 36. (Reproduced

by permission of Field Mark Publications.)



Methods Of Studying Probability ods of studying probability

Probability theory is concerned with determining the relationship between

the number of times some specific given event occurs and the number of

times any event occurs. For example, consider the flipping of a coin. One

might ask how many times a head will appear when a coin is flipped 100

times.

Determining probabilities can be done in two ways: theoretically and

empirically. The example of a coin toss helps illustrate the difference

between these two approaches. Using a theoretical approach, we reason that

in every flip there are two possibilities, a head or a tail. By assuming each

event is equally likely, the probability that the coin will end up heads is ½ or

0.5.

The empirical approach does not use assumptions of equal likelihood.

Instead, an actual coin flipping experiment is performed, and the number of

heads is counted. The probability is then equal to the number of heads

actually found divided by the total number of flips.



Basic Conceptsic concepts

Probability is always represented as a fraction, for example, the number of

times a "1 dot" turns up when a die is rolled (such as 1 out 6, or ⅙) or the

number of times a head will turn up when a penny is flipped (such as 1 out of

2, or ½). Thus the probability of any event always lies somewhere between 0

and 1. In this range, a probability of 0 means that there is no likelihood at all

of the given event's occurring. A probability of 1 means that the given event

is certain to occur.

Probabilities may or may not be dependent on each other. For example, we

might ask what is the probability of picking a red card OR a king from a deck

of cards. These events are independent because even if you pick a red card,

you could still pick a king.

As an example of a dependent probability (also called a conditional

probability), consider an experiment in which one is allowed to pick any ball

at random out of an urn that contains six red balls and six black balls. On the

first try, a person would have an equal probability of picking either a red or a black ball. The number of each color is the same. But the probability of

picking either color is different on the second try, since only five balls of one

color remain.



Applications Of Probability Theory theory

Probability theory was originally developed to help gamblers determine the

best bet to make in a given situation. Many gamblers still rely on probability

theory—either consciously or unconsciously—to make gambling decisions.

Probability theory today has a much broader range of applications than just

in gambling, however. For example, one of the great changes that took place

in physics during the 1920s was the realization that many events in nature

cannot be described with perfect certainty. The best one can do is to say

how likely the occurrence of a particular event might be.

When the nuclear model of the atom was first proposed, for example,

scientists felt confident that electrons traveled in very specific orbits around

the nucleus of the atom. Eventually they found that there was no basis for

this level of certainty. Instead, the best they could do was to specify the

probability that a given electron would appear in various regions of space in

the atom. If you have ever seen a picture of an atom in a science or chemistry

book, you know that the cloudlike appearance of the atom is a way of showing the probability that electrons occur in various parts of the atom.



Part 1 (b):

Theorical Probabilities and Empirical Probabilities

Theorical Probabilities:

Probability theory is the branch ofmathemati cs concerned withanalysisofrandom phenomena.[1] The central objects of probability theoryare randomvariables, stochastic processes, and events: mathematicalabstractions of non-deterministic events or measured quantities that may either besingle occurrences or

evolve over time in an apparently random fashion. Although an

individual coin toss or the roll of a dice is a random event, if repeated many times the sequence of random events will exhibit

certain statistical patterns, which can be studied and predicted.

Two representative mathematical results describing such patterns

are the law of large numbers and the central limit theorem.

As a mathematical foundation forstatistics, probability theory is

essential to many

human activities that involve quantitative analysis of large sets of

data. Methods of probability theory also apply to descriptions of complex systems

given only partial

knowledge of their state, as instatistical mechanics. A great

discovery of twentieth

centuryphysics was the probabilistic nature of physical phenomena

at atomic scales,

described in quantum mechanics.



Empirical Probabilities

Empirical probability, also known as relative frequency,or experimental

probability, is the ratio of the number favorable outcomes to thetotal number of trials, not in a sample space but in an actualsequence of experiments. In a more general sense, empiricalprobability estimates probabilities fromexperience andobservation.The phrase a posteriori probability has also been used as analternative to empirical probability or relative frequency. Thisunusual usage of the phrase is not directly related to Bayesianinference and not to be confused with its equally occasional use torefer to posterior probability, which is something else. In statistical

terms, the empirical probability is an estimate of a probability. If modellingusing a binomial distribution is appropriate, it isthemaximum likelihood estimate. It isthe Bayesian estimate for thesame case if certain assumptions are made for thepriordistribution of the probability

An advantage of estimating probabilities using empirical

probabilities is that this

procedure is relatively free of assumptions. For example, considerestimating the

probability among a population of men that they satisfy two

conditions: (i) that they are over 6 feet in height; (ii) that they

prefer strawberry jam to raspberry jam. A direct estimate could be

found by counting the number of men who satisfy both conditions

to give the empirical probability the combined condition. An

alternative estimate could be found by multiplying the proportion

of men who are over 6 feet in height with the proportion of menwho prefer strawberry jam to raspberry jam, but this estimate

relies on the assumption that the two conditions are statistically

independent.



A disadvantage in using empirical probabilities arises in estimating

probabilities which are either very close to zero, or very close to

one. In these cases very large Sample sizes would be needed in

order to estimate such probabilities to a good standard of relative

accuracy. Herestatistical models can help, depending on the

context, and in general one can hope that such models would

provide improvements in accuracy compared to empirical

probabilities, provided that the assumptions involved actually do

hold. For example, consider estimating the probability that the

lowest of the daily-maximum temperatures at a site in February in

any one year is less zero degrees Celsius. A record of such

temperatures in past years could be used to estimate this

probability. A model-based alternative would be to select of family

of probability distributions and fit it to the dataset contain pastyearly values: the fitted distribution would provide an alternative

estimate of the required probability. This alternative method can

provide an estimate of the probability even if all values in the

record are greater than zero.

Different between Empirical Probability & Theoretical Probability

Empirical probability is the probability a person calculates from

many different trials. For example someone can flip a coin 100

times and then record how many times it came up heads and how

many times it came up tails. The number of recorded heads

divided by 100 is the empirical probability that one gets heads.

The theoretical probability is the result that one should get if an

infinite number of trials were done. One would expect the

probability of heads to be 0.5 and the probability of tails to be 0.5

for a fair coin.



Part 3

Table 1 shows the sum of all dots on both turned-up when two dice aretossed simultaneously.

A)Complete Table 1 by listing all possible outcomes and theircorresponding probabilities.

Sum of the dots on

both turned-upfaces(x)

Possible

Outcomes

Probability,

P(x)

2 (1,1) 1/363 (1,2)(2,1) 2/364 (1,3)(3,1)(2,2) 3/36



5 (1,4)(4,1)(2,3)(3,2) 4/366 (1,5)(5,1)(2,4)(4,2)(3,3) 5/367 (1,6)(6,1)(2,5)(5,2)(3,4)(4,3) 6/368 (2,6)(6,2)(3,5)(5,3)(4,4) 5/369 (3,6)(6,3)(4,5)(5,4) 4/3610 (4,6)(6,4)(5,5) 3/36

11 (5,6)(6,5) 2/3612 (6,6) 1/36

(b)Based on Table 1 that you have competed, list all the possibleoutcomes of the following events and hence find their correspondingprobabilities:

A= {The two numbers are not the same}B= {The product of the two numbers is greater than 36}C= {Both numbers are prime or the difference between two

numbers is odd}D={The sum of the two numbers are even and both numbers are prime}

Solution

Part 3(b)

A={ (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2),(3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,5), (4,6),(5,1), (5,2), (5,3), (5,4),(5,6), (6,1), (6,2), (6,3), (6,4), (6,5)}P(A)=??

A’={(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}P(A’)=1/6

As P(A’)=P’(A)=1/6, thusP( A) =1- 1/6

B={},as the maximum product is 6X6=36. This event is impossible to occur.Thus,P(B)=0

Prime number(below six):2,3,5Odd number(below six):1,3,5

C = P U QC={(1,2), (1,4), (1,6), (2,1), (2,2), (2,3), (2,5), (3,2), (3,3), (3,4), (3,5), (3,6),



(4,1), (4,3), (4,5), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,3), (6,5)}=23/36

D = P ∩ R D={ (2,2), (3,3), (3,5), (5,3), (5,5)}P(D) =5/36

Answers:

A={ (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2),(3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,5), (4,6),(5,1), (5,2), (5,3), (5,4),(5,6), (6,1), (6,2), (6,3), (6,4), (6,5)}P(A)= 5/6

B={}

P(B)=0

C={(1,2), (1,4), (1,6), (2,1), (2,2), (2,3), (2,5), (3,2), (3,3), (3,4), (3,5), (3,6),(4,1), (4,3), (4,5), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,3), (6,5)}P(C)= 23/36

D={ (2,2), (3,3), (3,5), (5,3), (5,5)}P(D) =5/36

Part 4

Part 4(a)

a) Conduct an activity by tossing two dice simultaneously 50times. Observe thesum of all dots on both turned up faces.Complete the frequency table below.

Sum of the two

numbers( x )

Frequency(f )

fx

f x 2

2 2 4 8



3 4 12 364 4 16 645 9 45 2256 4 24 144

7 11 77 5398 4 32 2569 6 54 48610 3 30 30011 1 11 12112 2 24 288

total

From the table,

(i)

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(ii)

(iii)

Part 4(b,c)

Sum of thetwonumbers( x

)

Frequency(f )

fx fx 2

2 4 8 163 5 15 45

4 6 24 965 16 80 4006 12 72 4327 21 147 1029



8 10 80 6409 8 72 64810 9 90 90011 5 55 605

12 4 48 576total

From the table,

(i)

(ii)

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(iii)

Part 5

Part 5(a)

x 2 3 4 5 6 7 8 9 10 11 12P(x

)

1/3

6

1/1

8

1/1

2

1/9 5/3

6

1/6 5/3

6

1/9 1/1

2

1/1

8

1/3

6



Part 5(b)

b)Part 4 Part 5

n=50 N=100Mean 5.58 6.91 7.00

Variance 6.0436 6.1219 5.83StandardDeviation

2.456 2.474 2.415

We can see that, the mean, variance and standard deviation thatwe obtain through experiment in Part 4 are different but close tothe value in Part 5.

For the mean, when the number of trial increase from n=50 ton=100, its value get closer(from 6.58 to 6.91)to the theoreticalvalue. This is in accordance to the Law Of Large Number. We willdiscuss Law Of Large Number in next section.

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Nevertheless, the empirical variance and empirical standarddeviation that we obtain in Part 4 get further from the theoreticalvalue in Part 5. This violates The Law Of Large Number. This isprobably due to

a) The sample(n=100) is not large enough to see the change of value of mean,variance and standard deviation.

b) Law Of Large Number is not an absolute law.Violation of thislaw is possible though the probability is relative low.

In conclusion, the empirical mean, variance and standard deviationcan be different from the theoretical value. When the number of trial (number of sample) getting bigger, the empirical value shouldget closer to the theoretical value. However, violation of this rule

is still possible,especially when the number of trial (or sample) isnot large enough.

Part 5(c)

The range mean:-

Conjecture: As the number of toss, n, increases, the mean will get closer to 7. 7is the theoretical mean.

Image below support this conjecture where we can see that, after 500 toss, the

theoretical mean become very close to the theoretical mean, which is 3.5.



(Take note that this is experiment of tossing 1 die, but not 2 dice as what we do

in our experiment)

Average dice value again number of rolls

_______=average _______y=3.5

mean

value

100 200 300 400 500 600 700 800 900

1000

trial

FURTHER EXPLORATION

In probability theory, the law of large numbers (LLN) is a

theorem that

describes the result of performing the same experiment a large

number of times.

According to the law, the average of the results obtained from a

large number of trials should be close to the expected value, and

will tend to become closer as more trials are performed. For

example, a single roll of a six-sided die produces one of the



numbers 1, 2, 3, 4, 5, 6, each with equalprobability. Therefore, the

expected value of a single die roll is

According to the law of large numbers, if a large number of diceare rolled, the

average of their values (sometimes called the sample mean) is

likely to be close to 3.5, with the accuracy increasing as more dice

are rolled.

Similarly, when a fair coin is flipped once, the expected value

of the number of heads is equal to one half. Therefore, according

to the law of large numbers, the

proportion of heads in a large number of coin flips should be

roughly one half. In

particular, the proportion of heads after n flips will almost surely

converge to one half as napproaches infinity.

Though the proportion of heads (and tails) approaches half,

almost surely the

absolute (nominal) difference in the number of heads and tails will

become large as the number of flips becomes large. That is, theprobability that the absolute difference is a small number

approaches zero as number of flips becomes large. Also, almost

surely the ratio of the absolute difference to number of flips will

approach zero. Intuitively, expected absolute difference grows, but

at a slower rate than the number of flips, as the number of flips

grows.

The LLN is important because it "guarantees" stable long-

term results for random events. For example, while a casino may

lose money in a single spin of

the roulette wheel, its earnings will tend towards a predictable

percentage over a large number of spins. Any winning streak by a

player will eventually be overcome by the parameters of the game.

It is important to remember that the LLN only applies (as the name

indicates) when a large number of observations are considered.

There is no principle that a small number of observations willconverge to the expected value or that a streak of one value will

immediately be "balanced" by the others.

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