Transcript
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    PROJECT WORK FOR

    ADDITIONAL MATHEMATICS -3472-

    2010

    Probability in Our Life

    NAMA : SUHAIMI BIN ADININ

    ANGKA GILIRAN : HF018A022

    I/C NUMBER : 930902-12-6683

    SEKOLAH : XEA 2072

    S.M.K BATU SAPI

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    CONTENT

    No Contents Page

    1 Introduction 3

    2 Part 1 4

    3 Part 2 5

    4 Part 3 7

    5 Part 4 8

    6 Part 5 11

    7 Conclusion 19

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    INTRODUCTION

    Probability is a way of expressing knowledge or belief that an event will occur or has occurred.

    In mathematics the concept has been given an exact meaning in probability theory, that is used

    extensively in such areas of study as mathematics, statistics, finance, gambling, science, and

    philosophy to draw conclusions about the likelihood of potential events and the underlying

    mechanics of complex systems.

    Interpretations

    The wordprobabilitydoes not have a consistent direct definition. In fact, there are sixteen

    broad categories ofprobability interpretations, whose adherents possess different (and

    sometimes conflicting) views about the fundamental nature of probability:

    1. Frequentists talk about probabilities only when dealing with experiments that are

    random and well-defined. The probability of a random event denotes the relative

    frequency of occurrence of an experiment's outcome, when repeating the experiment.

    Frequentists consider probability to be the relative frequency "in the long run" ofoutcomes.

    [1]

    2. Bayesians, however, assign probabilities to any statement whatsoever, even when no

    random process is involved. Probability, for a Bayesian, is a way to represent an

    individual's degree of beliefin a statement, or an objective degree of rational belief,

    given the evidence.

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    Part 1

    a) The history of probability theory

    Cardano, the father of probability

    Probability theory originally was used for games of chance, or better known as

    gambling. A mathematician named Gerolamo Cardano used the theory of probability as

    a way to analyze games of chance, and better influence his luck in the sixteenth

    century. Yes, probability was used to gamble. Cardano is regarded as the father of

    probability as he wrote a book on that subject. His book, known as Liberde ludo

    aleae ("Book on Games ofChance"), was published in 1663, was the first book to deal

    with probability systematically, as well as teaching us how to cheat with probability.

    The branch of probability was further expanded upon by two mathematicians, namely

    Pierre de Fermat and Blaise Pascal in the seventeenth century. They conducted research

    on the problem of points, which is a mathematical problem concerning probability.

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    Blaise Pascal and Pierre de Fermat (Respectively). They built upon Gerolamo Cardanos

    work.

    In 1657, the field of probability was strengthened by Christiaan Huygens, who, under

    encouragement from Pascal, wrote a book on the subject, called De ratiociniis in ludo

    aleae ("On Reasoning in Games ofChance"), which he had published in 1657. This is

    regarded as the first book on probability theory, since it was published before Cardanos

    book.

    Later, the branch of probability evolved into modern probability theory with the help of

    Andrey Nikolaevich Kolmogorov, who laid the foundations of modern probability. He

    combined the sample space with the measure theory, creating his axiom system in

    1933. This became the standardized form of probability measurement, continuing until

    now.

    In our daily life, The Theory of Probability is used in various occurrences. One real life

    example is when we are playing cards, for example Blackjack. There are 52 cards in a

    deck. Each player receives two cards. The best pair of cards that one can receive is an

    Ace and one of the picture cards. Speaking in terms of probability, if there are three

    players, counting the banker, each player has a 6/169 chance of getting 21 points in the

    first deal.

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    Alternatively, the Theory of Probability is used for business. Probability is used in

    business to evaluate the risk in a decision. The higher probability of success ensures a

    lower chance of failing.

    Probability can help us understand the seemingly random forces and ensure a risk freeenvironment. Thus it is important for us to study probability.

    b) The difference between Theoretical Probability and Empirical Probability

    Theoretical probability is the branch of probability concerned with the theory. There is

    no concrete proof and all results are based only on calculation.

    Empirical probability, as its name suggests, is based on experiments and the results

    Mathematical treatment

    In mathematics, a probability of an event A is represented by a real number in the range from 0

    to 1 and written as P(A), p(A) or Pr(A).[6]

    An impossible event has a probability of 0, and a

    certain event has a probability of 1. However, the converses are not always true: probability 0

    events are not always impossible, nor probability 1 events certain. The rather subtle distinction

    between "certain" and "probability 1" is treated at greater length in the article on "almostsurely".

    The opposite or complementof an event A is the event [not A] (that is, the event ofA not

    occurring); its probability is given by P(not A) = 1 - P(A).[7]

    As an example, the chance of not

    rolling a six on a six-sided die is 1 (chance of rolling a six) . See

    Complementary event for a more complete treatment.

    If both the events A and B occur on a single performance of an experiment this is called the

    intersection or joint probability ofA and B, denoted as . If two events, A and B are

    independent then the joint probability is

    for example, if two coins are flipped the chance of both being heads is[8]

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    If either event A or event B or both events occur on a single performance of an experiment this

    is called the union of the events A and B denoted as . If two events are mutually

    exclusive then the probability of either occurring is

    For example, the chance of rolling a 1 or 2 on a six-sided die is

    If the events are not mutually exclusive then

    For example, when drawing a single card at random from a regular deck of cards, the chance of

    getting a heart or a face card (J,Q,K) (or one that is both) is , because of

    the 52 cards of a deck 13 are hearts, 12 are face cards, and 3 are both: here the possibilities

    included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards"

    but should only be counted once.

    Conditional probabilityis the probability of some event A, given the occurrence of some other

    event B. Conditional probability is written P(A|B), and is read "the probability ofA, given B". It is

    defined by

    [9]

    IfP(B) = 0 then is undefined.

    Summary of probabilities

    Event Probability

    A

    not A

    A or B

    A and B

    A given B

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    Theory

    Probability theory

    Like other theories, the theory of probability is a representation of probabilistic concepts in

    formal termsthat is, in terms that can be considered separately from their meaning. Theseformal terms are manipulated by the rules of mathematics and logic, and any results are then

    interpreted or translated back into the problem domain.

    There have been at least two successful attempts to formalize probability, namely the

    Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation (see probability

    space), sets are interpreted as events and probability itself as a measure on a class of sets. In

    Cox's theorem, probability is taken as a primitive (that is, not further analyzed) and the

    emphasis is on constructing a consistent assignment of probability values to propositions. In

    both cases, the laws of probability are the same, except for technical details.

    There are other methods for quantifying uncertainty, such as the Dempster-Shafer theory or

    possibility theory, but those are essentially different and not compatible with the laws of

    probability as they are usually understood.

    Empirical probability

    It also known as relative frequency, or experimental probability, is the ratio of the number

    favorable outcomes to the total number of trials,[1][2]

    not in a sample space but in an actual

    sequence of experiments. In a more general sense, empirical probability estimates probabilities

    from experience and observation.[3]

    The phrase a posteriori probability has also been used as

    an alternative to empirical probability or relative frequency.[4] This unusual usage of the phraseis not directly related to Bayesian inference and not to be confused with its equally occasional

    use to refer to posterior probability, which is something else.

    In statistical terms, the empirical probability is an estimate of a probability. If modelling using a

    binomial distribution is appropriate, it is the maximum likelihood estimate. It is the Bayesian

    estimate for the same case if certain assumptions are made for the prior distribution of the

    probability

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    Applications

    Two major applications of probability theory in everyday life are in risk assessment and in trade

    on commodity markets. Governments typically apply probabilistic methods in environmental

    regulation where it is called "pathway analysis", often measuring well-being using methods that

    are stochastic in nature, and choosing projects to undertake based on statistical analyses oftheir probable effect on the population as a whole.

    A good example is the effect of the perceived probability of any widespread Middle East conflict

    on oil prices - which have ripple effects in the economy as a whole. An assessment by a

    commodity trader that a war is more likely vs. less likely sends prices up or down, and signals

    other traders of that opinion. Accordingly, the probabilities are not assessed independently nor

    necessarily very rationally. The theory of behavioral finance emerged to describe the effect of

    such groupthink on pricing, on policy, and on peace and conflict.

    It can reasonably be said that the discovery of rigorous methods to assess and combineprobability assessments has had a profound effect on modern society. Accordingly, it may be of

    some importance to most citizens to understand how odds and probability assessments are

    made, and how they contribute to reputations and to decisions, especially in a democracy.

    Another significant application of probability theory in everyday life is reliability. Many

    consumer products, such as automobiles and consumer electronics, utilize reliability theory in

    the design of the product in order to reduce the probability of failure. The probability of failure

    may be closely associated with the product's warranty.

    Part 2

    a) List of all the possible outcomes when the die is tossed once= {1,2,3,4,5,6}

    b) List of all the possible outcomes when two dice are tossed simultaneously:

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    Part 3

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

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

    Sum of the dots on both

    turned-up faces (x)

    Possible outcomes Probability, P(x)

    2

    (1,1) 1/36

    3

    (1,2),(2,1) 1/18

    4

    (1,3),(2,2),(3,1) 1/12

    5

    (1,4),(2,3),(3,2),(4,1) 1/9

    6

    (1,5),(2,4),(3,3),(4,2),(5,1) 5/36

    7

    (1,6),(2,5),(3,4),(4,3),(5,2),

    (6,1)

    1/6

    8

    (6,2),(3,5),(4,4),(5,3),(6,3) 5/36

    9

    (3,6),(4,5),(5,4),(6,3) 1/9

    10

    (4,6),(5,5),(6,4) 1/12

    11

    (5,6),(6,5) 1/18

    12

    (6,6) 1/36

    Table 1

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    B) Based on table 1 that have I completed. All the possible outcomes of the following eventand their corresponding probabilities..

    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)

    B = { }

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

    (3,6),(4,1),(4,2),(4,3),(4,5),(5,4),(5,6),(6,1),(6,3),(6,5)

    D = {(2,2),(3,3),(3,5),(5,3),(5,5)}

    P(A)=5

    6

    P(B)=0

    P(C)=23

    36

    P(D)=5

    36

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    Part 4

    A) Conduct an activity by tossing two dice simultaneously 50 times. Observations

    sum of all dots on both turned-up faces. Complete the frequency table below.

    Sum of two numbers

    (x)

    Frequency FX X2 Fx2

    2 12 4 4

    3 39 9 27

    4 312 16 48

    5 735 25 175

    6 424 36 144

    7 856 49 392

    8 756 64 448

    9 872 81 648

    10 220 100 200

    11 5 55 121 605

    12 224 141 288

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    Table 2

    Based on the table 2 that I have completed, the value of

    (1)Mean =365

    50=7.3

    (2)Variance =2979

    50- 7.32 = 6.29

    (3)Standard deviation =2.508

    (B)Predict the value of the mean if the number of tosses is increased to 100 times. =Thevalue of the tossed will be increase.

    (c)Test your prediction in 9b) by continuing Activity 3(a) until the number of tosses is 100

    times.Then, determine the value of:

    VARIANCE = 2 - X2

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    SUM OF THE

    TWO

    NUMBERS(X)

    Frequency (x) F (x) X2 F(x2)

    2 2 4 4 8

    3 6 18 9 54

    4 8 32 16 128

    5 13 65 25 325

    6 13 78 36 468

    7 16 112 49 784

    8 1 4 112 64 894

    9 12 108 81 972

    10 10 100 100 1000

    11 4 44 121 984

    12 2 24 144 288

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    Mean =679

    100 = 6.79

    Variance =5470

    100-6.79

    Standard deviation =2.343

    Alternative metho

    ~ The tossing of dice may be simulated using the graphing calculator application programme.

    Part 5

    (A)

    Sum of the two

    number(x)

    P(x) xP(x) X2 X2P(x)

    2 1/36 2/36 4 4/36

    3 2/36 6/36 9 18/36

    4 3/36 12/36 16 48/36

    5 4/36 20/36 25 100/36

    6 5/36 30/36 36 180/36

    7 6/36 42 /36 49 294/36

    8 5/36 40/36 64 320/36

    9 4/36 36/36 81 324/36

    10 3/36 30/36 100 300/36

    11 2/36 22/36 121 242/36

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    12 1/36 12/36 144 144/36

    1994/36

    B)Compare the mean ,variance and the standard deviation obtain in part 4 and part 5.What can

    you say about the values? Explain in your own words your interpretation and your

    understanding of the values that you have obtained and related your answer to the Theorical

    and Emperical Probabilities.

    Part 4(A) Part 4(b) Part 5

    m 7.3 6.97 7

    v 6.29 5.489 5.833

    5d 2.508 2.343 2.415

    (c)If n is the number of times two dice are tossed simultaneously ,what is the range of the sum

    of all dots on the turned-up faces as n changes? Make your conjecture and support your

    conjecture.

    n mean

    50 7.3

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    100 6.79

    As n changes the range of mean of the sum of the range of mean is between 6 and 8.

    SUPPORTING CONJECTUR

    ~From the probability distribution graph, it shows that the range of mean is between 6 and 8.

    0

    0.05

    0.1

    0.15

    2 3 4 5 6 7

    Series 1

    Series 1

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    FURTHER EXPLORATION

    Law of large numbers

    Common intuition suggests that if a fair coin is tossed many times, then roughly half of the timeit will turn up heads, and the other half it will turn up tails. Furthermore, the more often the coinis tossed, the more likely it should be that the ratio of the number ofheads to the number oftails

    will approach unity. Modern probability provides a formal version of this intuitive idea, knownas the law of large numbers. This law is remarkable because it is nowhere assumed in the

    foundations of probability theory, but instead emerges out of these foundations as a theorem.Since it links theoretically derived probabilities to their actual frequency of occurrence in the real

    world, the law of large numbers is considered as a pillar in the history of statistical theory.[1]

    The law of large numbers (LLN) states that the sample average of (independent and

    identically distributed random variables with finite expectation ) converges towards thetheoretical expectation .

    It is in the different forms of convergence of random variables that separates the weakand thestronglaw of large numbers

    It follows from LLN that if an event of probability p is observed repeatedly during independent

    experiments, the ratio of the observed frequency of that event to the total number of repetitions

    converges towards p.

    Putting this in terms of random variables and LLN we have are independent Bernoulli

    random variables taking values 1 with probabilityp and 0 with probability 1-p. E(Yi) = p for all i

    and it follows from LLN that converges to p almost surely.

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    REFLECTION

    While I conducting the project, I had learned some

    moral values that I practice. This project had

    taught me to responsible on the works that are

    given to me to be completed. This project also had

    make me felt more confidence to do works and not

    to give up easily when we could not find the

    solution for the question. I also learned to be more

    discipline on time, which I was given about two

    weeks to complete these project and pass up to

    my teacher just in time. I also enjoy doing this

    project during my school holiday as I spend my

    time with friends to complete this project and it

    had tightens our friendship.