Topic outline

  • General

  • UNIT 1 :SEQUENCES AND SERIES

    Key unit competence: Apply arithmetic and geometric sequences to solve
                                                    problems in financial mathematics.

    1.0 INTRODUCTORY ACTIVITY
    Suppose that an insect population is growing in such a way that each new
    generation is 2 times as large as the previous generation. If there are 126
    insects in the first generation, on a piece of paper, write down the number
    of insects that will be there in second, third, fourth,…nthgeneration.

    How can we name the list of the number of insects for different
    generations?

    1.1 Generalities on sequences
    ACTIVITY 1.1
    Fold once an A4 paper, what is the fraction that represents the part
    you are seeing?
    Fold it twice, what is the fraction that represents the part you are
    seeing?
    What is the fraction that represents the part you are seeing if you
    fold it ten times?
    What is the fraction that represents the part you are seeing if you

    fold it n times?

    Write a list of the fractions obtained starting from the first until the

    nth fraction.

    C

    V

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    The empty sequence { } is included in most notions of sequences, but may
    be excluded depending on the context. Usually a numerical sequence is given

    by some formula egg nu     = f No , permitting to find any term of the sequence by its
    number n; this formula is called a general term formula.

    A second way of defining a sequence is to assign a value to the first (or the first
    few) term(s) and specify the nth term by a formula or equation that involves
    one or more of the terms preceding it. Sequences defined this way are said to

    be defined recursively, and the rule or formula is called a recursive formula.

    V

    Infinite and finite sequences
    Consider the sequence of odd numbers less than 11: This is 1, 3,5,7,9. This is
    a finite sequence as the list is limited and countable. However, the sequence
    made by all odd numbers is:
    1,3,5,7,9,...2n +1,...This suggests the definition that an infinite sequence is a
    sequence whose terms are infinite and its domain is the set of positive integers.
    Note that it is not always possible to give the numerical sequence by a general

    term formula; sometimes a sequence is given by description of its terms.

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    In each term, the numerator is the same as the term number, and the denominator

    is one greater the term number.

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    Common difference
    The fixed numbers that bind each sequence together are called the common
    differences.     Sometimes mathematicians use the letter d when referring to

    these types of sequences.

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    A

    1.5.Arithmetic Means of an arithmetic sequence
    ACTIVITY 1.5
    Suppose that you need to form an arithmetic sequence of 7 terms
    such that the first term is 2 and the seventh term is 20. Write down
    that sequence given that those terms are 2, A, B,C,D, E, 20 .
    If three or more than three numbers form an arithmetic sequence, then all terms

    lying between the first and the last numbers are called arithmetic means. If B

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    A

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    APPLICATION ACTIVITY 1.6
    1)Consider the arithmetic sequence 8, 12, 16, 20, … Find the
    expression for  Sn
    2) Sum the first twenty terms of the sequence 5, 9, 13,…
    3) The sum of the terms in the sequence 1, 8, 15, … is 396. How many
    terms does the sequence contain?
    4) Practical activity: A ceramic tile floor is designed in the shape
    of a trapezium 10m wide at the base and 5m wide at the top as

    illustrated on the figure bellow:

    D

    The tiles, 10cm by 10cm, are to be placed so that each successive
    row contains one less tile than the preceding row. How many tiles

    will be required?

    1.7 Harmonic sequences and its general term
    ACTIVITY 1.7
    Consider the following arithmetic sequence:
    2, 4, 6, 8, 10, 12, 14, 16, …2n,...
    a) Form another sequence whose terms are the reciprocals of the
    terms of the given sequence.
    b) What can you say about the new sequence? What is its first term,
    the third term and the general term? Is there a relationship between
    two consecutive terms?

    Harmonic sequence is a sequence of numbers in which the reciprocals of the

    terms are in arithmetic sequence. It is of the following form:

    X

    Remark
    To find the term of harmonic sequence, convert the sequence into arithmetic
    sequence then do the calculations using the arithmetic formulae. Then take
    the reciprocal of the answer in arithmetic sequence to get the correct term in

    harmonic sequence.

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    1. 8 Generalities on Geometric sequence and its general term
    ACTIVITY 1.8
    Take a piece of paper with a square shape.
    1. Cut it into two equal parts.
    2. Write down a fraction corresponding to one part according to
    the original piece of paper.
    3. Take one part obtained in step 2) and repeat step 1) and then step 2)
    4. Continue until you remain with a small piece of paper that you
    are not able to cut into two equal parts and write down the
    sequence of fractions obtained.
    5. Observe the sequence of numbers you obtained and give the

    relationship between any two consecutive numbers.

    Sequences of numbers that follow a pattern of multiplying a fixed number r

    from one term u1 to the next are called geometric sequences.

    The following sequences are geometric sequences:

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    1.9.Geometric Means
    ACTIVITY 1.9
    Suppose that you need to form a geometric sequence of 6 terms such
    that the first term is 1 and the sixth term is 243. Given that these

    terms are 1, A,B,C,D, 243 . Write down that sequence.

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    1. 10. Geometric series
    ACTIVITY 1.10
    During a competition of student teachers at the district level, 5 first
    winners were paid an amount of money in the way that the first got
    100,000Frw, the second earned the half of this money, the third got
    the half of the second’s money, and so on until the fifth who got the
    half of the fourth’s money.
    a) Discuss and calculate the money earned by each student from the
    second to the fifth.
    b) Determine the total amount of money for all the 5 student teachers.
    c) Compare the money for the first and the fifth student and discuss
    the importance of winning at the best place.
    d) Try to generalize the situation and guess the money for the
    student who passed at the nth place if more students were paid. In

    this case, evaluate the total amount of money for n students.

    d

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    1.12 Application of sequences in real life
    ACTIVITY 1.12
    Carry out a research in the library or on internet and find out at least
    3 problems or scenarios of the real life where sequences and series are applied.

    There are many applications of sequences. Sequences are useful in our daily lives
    as well as in higher mathematics. For example; the monthly payments made to
    pay off an automobile or home loan with interest portion, the list of maximum
    daily temperatures in one area for a month are sequences. Sequences are used
    in calculating interest, population growth, half-life and decay in radioactivity,
    etc.
    In economics and Finance, sequences and series can be used for example in
    solving problems related to:

    a) Final sum, the initial sum, the time period and the interest rate for an
    investment.
    The amount A after t years due to a principal P invested at an annual interest

    rate r compounded n times per year is

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    value at t = 0 ; r is the Interest rate expressed as a decimal; r is the number of
    years P is invested; A is the amount after t years.

    The amount A after t years due to a principal P invested at an annual interest

    rate r compounded continuously is

    d

    a

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    d

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    7) To save for her daughter’s college education, Martha decides to put $50
    aside every month in a bank guaranteed-interest account paying 4% interest
    compounded monthly.

    She begins this savings program when her daughter is 3 years old. How much
    will she have saved by the time she makes the 180th deposit? How old is her

    daughter at this time?

    t

    APPLICATION ACTIVITY 1.12
    1) If Linda deposits $1300 in a bank at 7% interest compounded
    annually, how much will be in the bank 17 years later?

    2) The population of a city in 1970 was 153,800. Assuming that
    the population increases continuously at a rate of 5% per year,
    predict the population of the city in the year 2000.

    3) To save for retirement, Manasseh, at age 35, decides to place
    2000Frw into an Individual Retirement Account (IRA) each year
    for the next 30 years. What will the value of the IRA be when
    Manasseh makes his 30th deposit? Assume that the rate of return
    of the IRA is 4% per annum compounded annually.

    4) A private school leader received permission to issue 4,000,000Frw
    in bonds to build a new high school. The leader is required to
    make payments every 6 months into a sinking fund paying 4%
    compounded semiannually. At the end of 12 years the bond

    obligation will be retired. What should each payment be?

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  • UNIT 2:LOGARITHMIC AND EXPONENTIAL EQUATIONS

    Key unit competence: Solve equations involving logarithms or exponentials

                                                 and apply them to model and solve related problems.

    2.0 INTRODUCTORY ACTIVITY

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    2.3 Application of exponential and logarithmic equations in real life
    2.3.1 Application of exponential equations to estimate the

    Population Growth

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    A population whose rate of decrease is proportional to the size of the population
    at any time obeys a law of the forms P = Ae−kt . The negative sign on exponent
    indicates that the population is decreasing. This is known as exponential
    decay.
    If a quantity has an exponential growth model, then the time required for it
    to double in size is called the doubling time. Similarly, if a quantity has an
    exponential decay model, then the time required for it to reduce in value by
    half is called the halving time. For radioactive elements, halving time is called

    half-life.

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    2.3.3 Application of logarithmic equations to determine the
    magnitude of an earthquake

    ACTIVITY 2.3.3
    Using internet and books, carry out a research and find out
    how logarithms can intervene to solve problems related to the

    determination of the magnitude of an earthquake.

    T

    Y

    Y

    APPLICATION ACTIVITY 2.3.3
    Earth quake can occur in any country. Assuming that there was an
    earthquake I*which occurred in Rwanda in a certain past year.
    Discuss how we can measure eventual earthquake which may occur
    in our country referring to I* instead of referring to earthquake that

    happened in western countries.

    2.3.4 Application of exponential equations on interest rate problems
    ACTIVITY 2.3.4
    Mr Cauchy has a rentable house for which he asked 20,000Frw at
    the first month. However, the client has pay at the beginning of every
    month by adding 1% of the money paid for the previous month. If
    the money is to be paid at the new bank account for Cauchy,

    a) Calculate the money kept on Cauchy’s account in the middle of
    the second, the third and the fourth month.
    b) What is the type of sequence made by the money to be paid by
    Cauchy’s client? Determine its general term.
    c) Discuss the formula to be used to find the money Mr Cauchy

    will find on his account at the end of 12 months.

    U

    Example
    1) Mr. John operates an account with a certain bank which pays a compound
    interest rate of 13.5% per annum. He opened the account at the beginning of
    the year with 500,000 FRW and deposits the same amount of money at the
    beginning of every year. Calculate how much he will receive at the end of 9
    years.

    After how long will the money have accumulated to 3.32 million of Rwandan

    Fracs?

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    R

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    Hence it will take 4.6 years for the amount to accumulate to 3.32 million FRW

    2) A man deposits 800,000 FRW into his savings account on which interest is
    15% per annum. If he makes no withdrawals, after how many years will his

    balance exceed 8 million FRW?

    6

    APPLICATION ACTIVITY 2.3.4
    What annual rate of interest compounded annually should you seek

    if you want to double your investment in 5 years?

    2.3.5 Application of exponential equations to determine the
    mortgage payments

    ACTIVITY 2.3.5
    A loan with a fixed rate of interest is said to be amortized if both
    principal and interest are paid by a sequence of equal payments

    made over equal periods of time.

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    U

    Examples:
    1) Mr. Clement has just purchased a radio of 300,000Frw and has made a down
    payment of 60,000Frw. He can amortize the balance (300,000Frw-60,000Frw)
    at 6% for 30 years.
    (a) What are the monthly payments?
    (b) What is his total interest payment?
    (c) After 20 years, what equity does he have in his radio (that is, what is the

    sum of the down payment and the amount paid on the loan)?

    E

    2) When Mr. Thomas Rwambikana died, he left an inheritance of 15,000Frw
    for his family to be paid to them over a 10-year period in equal amounts at the
    end of each year. If the 15,000Frw is

    invested at 4% per annum, what is the annual payout to the family?

    Solution:
    This example asks what annual payment is needed at 4% for 10 years to disperse
    15,000Frw. That is, we can think of the 15,000Frw as a loan amortized at 4%
    for 10 years. Thepayment needed to pay off the loan is the yearly amount Mr.
    Rwambikana’s family willreceive.

    The yearly payout P is

    C

    3) Mr Unen is 20 years away from retiring and starts saving $100 a month in
    an accountpaying 6% compounded monthly. When he retires, he wishes to
    withdraw a fixedamount each month for 25 years. What will this fixed amount

    be?

    H

    APPLICATION ACTIVITY 2.3.5
    A corporation is faced with a choice between two machines, both of
    which are designedto improve operations by saving on labor costs.
    Machine A costs $8000 and will generatean annual labor savings of
    $2000. Machine B costs $6000 and will save $1800 in laborannually.
    Machine A has a useful life of 7 years while machine B has a useful
    life of only5 years. Assuming that the time value of money (the
    investment opportunity rate) of the corporation is 10% per annum,
    which machine is preferable? (Assume annual compounding and

    that the savings is realized at the end of each year).

    G

    N

  • UNIT 3:ACQUAINTED WITH TEACHING AND LEARNING

    Key unit competence: Solve problem involving the system of linear equations

                                                  using matrices.

    3.0 INTRODUCTORY ACTIVITY

    A Farmer Kalisa bought in Ruhango Market 5 Cocks and 4 Rabbits and
    he paid 35,000Frw, on the following day, he bought in the same Market 3
    Cocks and 6 Rabbits and he paid 30,000Frw.

    a)Arrange what Kalisa bought according to their types in a simple table

    as follows

    F

    b)Discuss and explain in your own words how you can determine the cost
    of 1 Cocks and 1 Rabbit.

    3.0 INTRODUCTORY ACTIVITY
    Matrices provide a means of storing large quantities of information in such a
    way that each piece can be easily identified and manipulated. They facilitate the
    solution of large systems of linear equations to be carried out in a logical and
    formal way so that computer implementation follows naturally. Applications of
    matrices extend over many areas of engineering including electrical network

    analysis and robotics.

    3.1. Definition and order of matrix

    ACTIVITY 3.1
    1) One shop sold 20 cell phones and 31 computers in a particular
    month. Another shop sold 45 cell phones and 23 computers in
    the same month. Present this information as an array of rows and columns.

    2) a) Observe and complete the number of students in the year two

    classes on one Monday.

    D

    b) If every class gets new students on Tuesday such that in SME they
    have 2 boys and 1 girls, in SSE they receive 1 girl and 1 boy, Complete
    the table for new students.

    c) Complete the table for all students in an array of rows and columns.

    CONTENT SUMMARY
    A matrix is a rectangular arrangement of numbers or algebraic expressions which
    illustrate the data for a real life model in rows and columns. A matrix is denoted
    with a capital letter: A,B,C,…and the elements are enclosed by parenthesis

    eggor square brackets[ ].

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    Types of matrices
    There are several types of matrices, but the most commonly used are

    1) Row matrix: matrix formed by one row

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    3.2. Operations on matrices
    3.2.1 Addition and subtraction of matrices

    ACTIVITY 3.2.1
    1) In a survey of 900 people, the following information was obtained:
    200 males thought federal defense spending was too high,150 males
    thought federal defense spending was too low, 45 males had no
    opinion, 315 females thought federal defense spending was too high
    125 females thought federal defense spending was too low, 65
    females had no opinion.

    Discuss and arrange these data in a rectangular array as follows:

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    3.3.2 Multiplying matrices
    ACTIVITY 3.2.2
    1) A clothing store sells men’s shirts for $40, silk ties for $20, and
    wool suits for $400.
    Last month, the store had sales consisting of 100 shirts, 200 ties, and
    50 suits.
    Using matrix, discuss and explain in your own words how to

    determine the total revenue due to these sales.

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    CONTENT SUMMARY

    Let A, B,C be matrices of order two or three

    1) Associative

    A×(B×C) = ( A× B)×C

    2) Multiplicative Identity
    A× I = A, where I is the identity matrix with the same order as matrix A.
    3) Not Commutative
    A× B ≠ B× A
    4) Distributive

    A ×(B +C) = ( A× B) + ( A×C)

    f

    Find
    a) The product A× B
    b) The product B× A

    c) Conclude about the commutativity of multiplication of matrices

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    Observation: The given matrices commute in multiplication.
    Notice
    • If AB = 0, it does not necessarily follow that A = 0 or B = 0 .
    • Commuting matrices in multiplication:In general the multiplication of
    matrices is not commutative, i.e, AB ≠ BA , but we can have the case where
    two matrices A and B satisfy AB = BA. In this case A and B are said to be
    commuting.

    Trace of matrix
    The sum of the entries on the leading diagonal of a square matrix, A, is known

    as the trace of that matrix, notedtr ( A) .

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    3.4. Determinants and inverse of a matrix of order two and three

    3.4.1. Determinant of order two or three

    ACTIVITY 3.4.1

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    CONTENT SUMMARY

    Consider two matrices, one of order two and another one of order three:

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    The determinant of A is calculated by SARRUS rule:
    The terms with a positive sign are formed by the elements of the principal
    diagonal and those of the parallel diagonals with its corresponding opposite
    vertex.
    The terms with a negative sign are formed by the elements of the secondary
    diagonal and those of the parallel diagonals with its corresponding opposite

    vertex.

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    3.5. Applications of matrices and determinants
    3.5.1 Solving System of linear equations using inverse matrix
    ACTIVITY 3.5.1
    A Farmer Kalisa bought in Ruhango Market 5 Cocks and 4 Rabbits
    and he paid 35,000Frw, on the following day, he bought in the same
    Market 3 Cocks and 6 Rabbits and he paid 30,000Frw.
    a) Considering x as the cost for one cock and y the cost of one
    Rabbit, formulate equations that illustrate the activity of Kalisa;

    b) Make a matrix A indicating the number of cocks and rabbits

    gt

    v

    Notice
    • If at least, one of bi is different of zero the system is said to be non-homogeneous
    and if all bi are zero the system is said to be homogeneous.

    • The set of values of x, y, z that satisfy all the equations of system (1) is
    called solution of the system.

    • For the homogeneous system, the solution x = y = z = 0 is     called trivial
    solution    . Other solutions are non-trivial solutions.
    Non- homogeneous system cannot have a trivial solution as at least one of

    x, y, z is not zero.

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    Note: The method of GAUSS helps us to solve the above system where CRAMER’S

    method cannot.

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  • UNIT 4 : BIVARIATE STATISTICS

     Key unit Competence: Extend understanding, analysis and interpretation
                                                  of bivariate data to correlation coefficients and

                                                  regression lines

    4.0 INTRODUCTORY ACTIVITY

    In Kabeza village, after her 9 observations about farming,
    UMULISA saw that in every house observed, where there is a cow (X) if
    there is also domestic duck (Y), then she got the following results:
    (1,4) ,( 2,8) , (3,4) , (4,12) , (5,10),(6,14) , (7,16) , (8,6 ), (9,18)

    a. Represent this
    information graphically in (x, y) − coordinates .
    b. Find the equation of line joining any two points of the graph and guess the
    name of     this line.

    c. According to your observation     from (a), explain in your own words if there is any
    relationship between the variation of Cows (X) and the variation of domestic duck (Y).
    4
    4.1 Bivariate data, scatter diagram and types of correlation
    ACTIVITY 4.1
    Consider the situation in which the mass, y (g), of a chemical is
    related to the time , x minutes, for which the chemical reaction has

    been taking place ,according to the table.

    R

    a) Plot the above information in (x, y) coordinates.

    b) Explain in your own words the relationship between x and y

    In statistics, bivariate or double series includes technique of analyzing data in
    two variables, when focus on the relationship between a dependent variable-y
    and an independent variable-x.

    For example, between age and weight, weight and height, years of education
    and salary, amount of daily exercise and cholesterol level, etc. As with data for a
    single variable, we can describe bivariate data both graphically and numerically.
    In both cases we will be primarily concerned with determining whether there
    is a linear relationship between the two variables under consideration or not.

    It should be kept in mind that a statistical relationship between two variables
    does not necessarily imply a causal relationship between them. For example,
    a strong relationship between weight and height does not imply that either
    variable causes the other.

    Scatter plots or Scatter diagram and types of correlation
    Consider the following data which relate x, the respective number of branches
    that 10 different banks have in a given common market, with y, the corresponding

    market share of total deposits held by the banks:

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    The scatter plot or scatter diagram (in the figure above) indicates that, roughly
    speaking, the market share increases as the number of branches increases. We
    say that x and y have a positive correlation.

    On the other hand, consider the data below, which relate average daily
    temperature x, in degrees Fahrenheit, and daily natural gas consumption y, in

    cubic metre.

    E

    Finally, consider the data items (x, y) below, which relate daily temperature x

    over a 10-day period to the Dow Jones stock average y.

    E
    We see that y tends to decrease as x increases. Here, x and y have a negative
    correlation.
    Finally, consider the data items (x, y) below, which relate daily temperature x
    over a 10-day period to the Dow Jones stock average y: (63, 3385); (72, 3330);
    (76, 3325); (70, 3320); (71, 3330); (65, 3325); (70, 3280); (74, 3280) ;(68,

    3300); (61, 3265).

    4

    There is no apparent relationship between x and y (no correlation or Weak

    correlation.

    APPLICATION ACTIVITY 4.1
    One measure of personal fitness is the time taken for an individual’s
    pulse rate to return to normal after strenuous exercise, the greater the
    fitness, the shorter the time. Following a short program of strenuous
    exercise Norman recorded his pulse rates P at time t minutes after
    he had stopped exercising. Norman’s results are given in the table

    below.

    3

    a) Draw a scatter diagram to represent this information in
    (x, y)coordinates

    b) Explain the relationship between Norman’s pulse P and time t.

    R

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    In case of two variables, say x and y, there is another important result called
    covariance of x and y, denoted cov(x, y) .
    The covariance of variables x and y is a measure of how these two variables
    change together. If the greater values of one variable mainly correspond with the
    greater values of the other variable, and the same holds for the smaller values,
    i.e. the variables tend to show similar behavior, the covariance is positive. In
    the opposite case, when the greater values of one variable mainly correspond
    to the smaller values of the other, i.e. the variables tend to show opposite
    behavior, the covariance is negative. If covariance is zero the variables are said
    to be uncorrelated, itmeans that there is no linear relationship between them.

    Therefore, the sign of covariance shows the tendency in the linear relationship

    between the variables. The magnitude of covariance is not easy to interpret.

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    The Pearson’s coefficient of correlation (or Product moment coefficient of
    correlation or simply coefficient of correlation), denoted by r, is a measure of

    the strength of linear relationship between two variables.

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    Properties of the coefficient of correlation
    a) The coefficient of correlation does not change the measurement scale.
    That is, if the height is expressed in meters or feet, the coefficient of
    correlation does not change.

    b) The sign of the coefficient of correlation is the same as the covariance.

    c) The square of the coefficient of correlation is equal to the product of the
    gradient of the regression line of y on x , and the gradient of the regression

    line of x on y .

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    g) If the linear coefficient of correlation takes values closer to −1, the
    correlation is strong and negative, and will become stronger the closer
    rapproaches −1.

    h) If the linear coefficient of correlationtakes values close to     1 the correlation
    is strong and positive, and will become stronger the closer r approaches 1

    i) If the linear coefficient of correlationtakes values close to     0, the correlation is weak.

    j) If     r = 1or r = −1, there is perfect correlation and the line on the scatter
    plot is increasing or decreasing respectively.

    k) If r = 0, there is no linear correlation.

    Examples:
    1) A test is made over 200 families on number of children (x) and number of
    beds y per family. Results are collected in the table below

    M

    a) What is the average number for children and beds per a family?
    b) Find the covariance.
    c) Can we confirm that there is a high linear correlation between the number of
    children and number of beds per family?

    Solution
    a) Average number of children per family:

    Contingency table:

    M

    M

    N

    M

    M

    Spearman’s coefficient of rank correlation
    A Spearman coefficient of rank correlation or Spearman’s rho is measure
    of statistical dependence between two variables. It assesses how well the
    relationship between two variables can be described using a monotonic
    function. The Spearman’s coefficient of rank correlation is denoted and defined by
    N
    Where, d refers to the difference of ranks between paired items in two series and
    n is the number of observations. It is much easier to calculate the Spearman’s
    coefficient of rank correlation than to calculate the Pearson’s coefficient
    of correlation as there is far less working involved. However, in general, the
    Pearson’s coefficient of correlation is a more accurate measure of correlation
    when data are numerical.

    Method of ranking
    Ranking can be done in ascending order or descending order.

    Examples:
    1) Suppose that we have the marks, x, of seven students in this order:
    12, 18, 10, 13, 15, 16, 9
    We assign the rank 1, 2, 3, 4, 5, 6, 7 such that the smallest value of x will be
    ranked 1.

    That is

    N

    F

    D

    T

    H

    T

    E


    CONTENT SUMMARY
    We use the regression line of y on x to predict a value of y for any given value
    of x and vice versa, we use the regression line of x on y, to predict a value of
    x for a given value of y. The “best” line would make the best predictions: the
    observed y-values should stray as little as possible from the line. This straight
    line is the regression line from which we can adjust its algebraic expressions

    and it is written as y = ax + b , where a is the gradient and b is the y-intercept.

    F

    Z

    M

    F

    R

    R

    R

    APPLICATION ACTIVITY 4.4

    1. Consider the following table

    m

    4.5 Interpretation of statistical data (Application)
    ACTIVITY 4.5
    Explain in your own words how statistics, especially bivariate
    statistics, can be used in our daily life.

    Bivariate statistics can help in prediction of a value for one variable if we know
    the value of the other.

    Examples:
    1. One measure of personal fitness is the time taken for an individual’s pulse
    rate to return to normal after strenuous exercise, the greater the fitness, the
    shorter the time. Following a short program of strenuous exercise Norman
    recorded his pulse rates P at time t minutes after he had stopped exercising.

    Norman’s results are given in the table below.

    d

    d

    d

    g

    iv. State with a reason whether it is sensible to conclude from your
    answer to part( iii) that and are linearly related.
    v. The line of regression of on x has equation y = ax + b . Calculate the
    value of a and b each correct to three significant figures.
    vi. Use your regression line to estimate what the contrast index
    corresponding to the damaged piece of film would have been if the
    piece has been undamaged.
    vii.State with a reason, whether it would be sensible to use your
    regression equation to estimate the contrast index when the quantity

    of chemical applied to the film is zero.

    4.6 END UNIT ASSESSMENT
    1) The following results were obtained from lineups in Mathematics

    and Physics examinations:

    4

    f

    4) The table below shows the marks awarded to six students in a

    competition:

    d

    Calculate a coefficient of rank correlation.
    5) A company is to replace its fleet of cars. Eight possible models
    are considered and the transport manager is asked to rank them,
    from 1 to 8, in order of preference. A saleswoman is asked to use
    each type of car for a week and grade them according to their

    suitability for the job (A-very suitable to E-unsuitable).

    f

    a. Calculate the Spearman’s coefficient of rank correlation between
    i. price and transport manager’s rankings,
    ii. price and saleswoman’s grades.
    b. Based on the result of a. state, giving a reason, whether it would
    be necessary to use all three different methods of assessing the cars.
    c. A new employee is asked to collect further data and to do some
    calculations. He produces the following results:
    The coefficient of correlationbetween
          i. price and boot capacity is 1.2,
          ii. maximum speed and fuel consumption in miles per
               gallons is -0.7,
          iii. price and engine capacity is -0.9
    For each of his results say, giving a reason, whether you think
    it is reasonable.

    d. Suggest two sets of circumstances where Spearman’s coefficient
    of rank correlation would be preferred to the Pearson’s

    coefficient of correlation as a measure of association.

  • UNIT 5: CONDITIONAL PROBABILITY AND BAYES THEOREM

    Key unit Competence: Apply rules of probability to solve problems related

                                                  to dependent and independent events.

    5.0 INTRODUCTORY ACTIVITY

    1) Consider a machine which manufactures electronic components. These
    must meet certain specification. The quality control departmentregularly
    samples the components.
    Suppose, on average, 92 out of 100 components meet the specification.
    Imaginethata
    Componentisselected at random and let A be the outcome that a component
    meets the specification; let B be the outcome that a component does not
    meet the specification.
    a) Explain in your own words and determine the probability that a
    components meet the specification.
    b) Explain in your own words and determine the probability that a
    component does not meet the specification.
    2) A box contains 4 whitechalks and 3 black chalks. One chalk is drawn at
    random; its color is noted but not replaced in the box.
    a) What is the probability of selecting2 white chalks?

    b) Determine the probability of selecting 3 white and 2 black chalks.

    Probability is a measure of the likelihood of the occurrence of a particular

    outcome.

    5.1Tree diagram
    ACTIVITY 5.1
    A box contains 4 blue pens and 6 black pens. One pen is drawn at
    random, its color is noted and the pen is replaced in the box. A pen is
    again drawn from the box and its color is noted.
    1) For the 1st trial, what is the probability of choosing a blue pen
    and probability of choosing a black pen?
    2) For the 2nd trial, what is the probability of choosing a blue pen
    and probability of choosing a black pen? Remember that after
    the 1st trial the pen is replaced in the box.
    3) In the following figure complete the missing colors and

    probabilities

    W

    CONTENT SUMMARY
    A tree diagram is one way of illustrating the probabilities of certain outcomes
    occurring when two or more trials take place in succession by use of arrows in
    the form of a tree and branches. A tree diagram has branches and sub-branches
    which help us to see the sequence of events and all the possible outcomes at

    each stage.

    The outcome is written at the end of the branch and the fraction on the branch
    gives the probability of the outcome occurring.

    For each     trial the number of branches is equal to the number of possible
    outcomes of that trial.

    Examples:
    1) A bag contains 8 balls of which 3 are red and 5 are green. One ball is drawn at
    random, its colour is noted and the ball replaced in the bag. A ball is again drawn
    from the bag and its colour is noted. Find the probability that the 2 balls drawn
    will be
    a) red followed by green,
    b) red and green in any order,

    c) of the same colour.

    M

    M

    M

    APPLICATION ACTIVITY 5.1
    1. Calculate the probability of three coins landing on: Three heads.
    2. A class consists of six girls and 10 boys. If a committee of three is
    chosen at random, find the probability of:
    a) Three boys being chosen.
    b) Exactly two boys and a girl being chosen.
    c) Exactly two girls and a boy being chosen.
    d) Three girls being chosen.
    3. A bag contains 7 discs, 2 of which are red and 5 are green. Two discs
    are removed at random and their colors noted. The first disk is not
    replaced before the second is selected. Find the probability that
    the discs will be:
    a) both red    b) of different colors      c) the same colors.
    4. Three discs are chosen at random, and without replacement, from a
    bag containing 3 red, 8 blue and 7 white discs. Find the probability
    that the discs chosen will be

    a) all red      b) all blue     c) one of each color.

    5.2The Addition law of probability
    ACTIVITY 5.2
    Consider a machine which manufactures car components. Suppose
    each component falls into one of four categories:top quality,
    standard, substandard, reject
    After many samples have been taken and tested, it is found that
    under certain specific conditions the probability that a component

    falls into a category is as shown in the following table.

    M

    M

    M

    M

    D

    M

    5.3 Independent events
    ACTIVITY 5.3
    A box contains 3 red pens, 4 green pens and 5 blue pens. One pen is
    taken from the box and then replaced. Another pen is taken from the
    box. Let A be the event “the first pen is red” and B be the event the
    second pen is blue.”
    Is the occurrence of event B affected by the occurrence of event A?

    Explain.

    S

    G

    G

    W

    D

    5.4.Dependent events
    ACTIVITY 5.4
    Suppose that you have a deck of 52 cards. You can draw a card from
    that deck , continue without replacing it, and then draw a second card .
    a) What is the sample space for each event?
    b) Suppose you select successively two cards, what is the probability
    of selecting two red cards?
    c) Explain if there is any relationship (Independence or dependence)
    between those two events considering the sample space. Does the
    selection of the first card affect the selection of the second card?
    When the outcome or occurrence of the first event affects the outcome or
    occurrence of the second event in such a way that the probability is changed,
    the events are said to be dependent.

    Examples:
    1)Suppose a card is drawn from a deck and not replaced, and then the second
    card is drawn. What is the probability of selecting an ace on the first card and a

    king on the second card?

    F

    Note that:
    The event of getting a king on the second draw given that an ace was drawn the
    first time is called a conditional probability.

    APPLICATION ACTIVITY 5.4
    The world wide Insurance Company found that 53% of the residents
    of a city had home owner’s Insurance with its company of the clients,
    27% also had automobile Insurance with the company. If a resident
    is selected at random, find the probability that the resident has
    both home owner’s and automobile Insurance with the world wide

    Insurance Company.

    5.5 Conditional probability
    ACTIVITY 5.5
    A box contains 3 red pens, 4 green pens and 5 blue pens. One pen is
    taken from the box and is not replaced. Another pen is taken from
    the box. Let A be the event “the first pen is red” and B be the event
    “the second pen is blue.”
    Is the occurrence of event B affected by the occurrence of event A?

    Explain.

    F

    Examples:
    1) A die is tossed. Find the probability that the number obtained is a 4 given

    that the number is greater than 2.

    E

    2) At a middle school, 18% of all students play football and basketball, and 32%
    of all students play football. What is the probability that a student who plays

    football also plays basketball?

    F

    Notice:
    Contingency table
    Contingency table (or Two-Way table) provides a different way of calculating
    probabilities. The table helps in determining conditional probabilities quite
    easily. The table displays sample values in relation to two different variables
    that may be dependent or contingent on one another.
    Below, the contingency table shows the favorite leisure activities for 50 adults,
    20 men and 30 women. Because entries in the table are frequency counts, the

    table is a frequency table.

    4

    Calculate the following probabilities using the table:
    a) P(person is a car phone user)
    b) P(person had no violation in the last year)
    c) P(person had no violation in the last year AND was a car phone user)
    d) P(person is a car phone user OR person had no violation in the last year)
    e) P(person is a car phone user GIVEN person had a violation in the last year)
    f) P(person had no violation last year GIVEN person was not a car phone user)
    D

    Z
    b. The respondent was a male, given that the respondent answered no.
    M
    M

    M
    APPLICATION ACTIVITY 5.5
    The world wide Insurance Company found that 53% of the residents
    of a city had home owner’s Insurance with its company of the clients,
    27% also had automobile Insurance with the company. If a resident
    is selected at random, find the probability that the resident has
    both home owner’s and automobile Insurance with the world wide
    Insurance Company.

    1. A jar contains black and white marbles. Two marbles are chosen without
    replacement. The probability of selecting a black marble and then a white
    marble is 0.34, and the probability of selecting a black marble on the first
    draw is 0.47. What is the probability of selecting a white marble on the
    second draw, given that the first marble drawn was black?

    2. A bag contains five discs, three of which are red. A box is contains six discs,
    four of which are red. A card is selected at random from a normal pack of
    52 cards, if the card is a club a disc is removed from the bag and if the card
    is not a club a disc is removed from the box. Find the probability that, if the
    removed disc is red it came from the bag.

    3. The probability that Gerald parks in a no-parking zone and gets a parking
    ticket is 0.06, and the probability that Gerald cannot find a legal parking
    space and has to park in the no-parking zone is 0.20. On Tuesday, Gerald
    arrives at Headquarter office and has to park in a no-parking zone. Find the

    probability that he will get a parking ticket.

    C

    X

    S

    D

    S#

    APPLICATION ACTIVITY 5.6
    1. 20% of a company’s employees are engineers and 20% are
    economists. 75% of the engineers and 50% of the economists hold
    a managerial position, while only 20% of non-engineers and noneconomists
    have a similar position. What is the probability that
    an employee selected at random will be both an engineer and a
    manager?

    2. The probability of having an accident in a factory that triggers
    an alarm is 0.1. The probability of its sounding after the event
    of an incident is 0.97 and the probability of it sounding after no
    incident has occurred is 0.02. In an event where the alarm has
    been triggered, what is the probability that there has been no

    accident?

    5.7 END UNIT ASSESSMENT
    1) The probability that it is Friday and that a student is absent is
    0.03. Since there are 5 school days in a week, the probability that
    it is Friday is 0.2. What is the probability that a student is absent
    given that today is Friday?

    2) Dr. Richard is conducting a survey of families with 3 children.
    If a family is selected at random, what is the probability that
    the family will have exactly 2 boys if the second child is a boy?
    Assume that the probability of giving birth to a boy is equal to the
    probability of giving birth to a girl.

    3) A 12-sided die (dodecahedron) has the numerals 1 through 12 on
    its faces. The die is rolled once, and the number on the top face is
    recorded. What is the probability that the number is a multiple of
    4 if it is known that it is even?

    4) At Kennedy Middle School, the probability that a student takes
    Technology and Spanish is 0.087. The probability that a student
    takes Technology is 0.68. What is the probability that a student
    takes Spanish given that the student is taking Technology?

    5) A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn
    at random. What is the probability that none of the balls drawn

    is blue?

    6) In a certain college, 5% of the men and 1% of the women are taller
    than 180 cm. Also, 60% of the students are women. If a student is
    selected at random and found to be taller than 180 cm, what is the

    probability that this student is a woman?

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  • Topic 6

  • Topic 7

  • Topic 8

  • Topic 9