Topic outline

  • UNIT1:Trigonometric Formulae and Equations

    Introductory activity

    The height h of the cathedral is 485 m. The angle of elevation 
    of the top of the Cathedral from a point 280 m away from the 
    base of its steeple on level ground is θ . By using trigonometric 

    concepts, find the value of this angle θ in degree.

    C

    Trigonometry studies relationship involving lengths and angles 
    of a triangle. The techniques in trigonometry are used for 
    finding relevance in navigation particularly satellite systems 
    and astronomy, naval and aviation industries, land surveying, in 
    cartography and (creation of maps) and medicine. Even if those 
    are the scientific applications of the concepts in trigonometry, 
    most of the mathematics we study would seem to have little reallife application.
     Trigonometry is really relevant in our day to day 

    activities.

    Objectives

    By the end of this unit, a student will be able to:
    ᇢ Solve trigonometric equations.
    ᇢ Use trigonometric formulae and equations in 

    real life.

    1.1. Trigonometric formulae

    Activity 1.1

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    Deduce the formula of
    tan(A B + − ),sin(A B),cos(A B − − ), tan(A B)
    Deduce the formula of

    tan(A B + − ),sin(A B),cos(A B − − ), tan(A B)

    1.1.1. Addition and subtraction formulae

    From activity 1.1:

    The addition and subtraction formulae are

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    Addition and subtraction formulae are useful for finding 

    trigonometric number of some angles.

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    1.1.2. Double angle formulae

    Activity 1.2

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    1.1.3. Half angle formulae

    Activity 1.3

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    Solution

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    Solution

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    Example 1.10

    Transform in product the sum cos 2 cos 4 x x

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    Activity 1.6

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    Activity 1.9

    Discuss how trigonometric theory is used in refraction of light.
    In optics, light changes speed as it moves from one medium to 
    another (for example, from air into the glass of the prism). This 
    speed change causes the light to be refracted and to enter the new 
    medium at a different angle (Huygens principle). 
    A prism is a transparent optical element with flat, polished surfaces 

    that refract light. 

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    1.3.3. 1.3.3 Application in medicine
    Activity 1.10
    Conduct research in different books of the library or on the 
    internet to discover the application of trigonometry in the field 
    of medicine such as heart’s electrical activities, trucking the 
    rhythms of lungs capacity, testing electrical activities of the 
    brain and abnormalities in the brain...
    Trigonometric concepts such as sinusoidal waves contribute to 
    various medical testing and interpretation of those test results. 
    1. Electrocardiography: The measurement of electrical activities 
    in the heart. Through this process, it is possible to determine 
    how long the electrical wave takes to travel from one part of 
    the heart to the next by showing if the electrical activity is 
    normal or slow, fast or irregular. 
    2. Pulmonary function testing: a spirometer is used to measure 
    the volume of air inhaled and exhaled while breathing by 
    recording the changing volume over time. The output of a 
    spirogram can be quantified using trigonometric equations 
    and generally, it is possible to describe any repeating rhythms 

    by a sine wave:

    S

    In humans, when each breath is completed, the lung still 
    contains a volume of air, called the functional residual capacity 
    (approximately 2200 mL in humans). Each inhalation adds from 
    500 mL of additional air for normal (resting) breathing. Each 
    exhalation removes approximately the same volume as was 
    inhaled. The volume of air inhaled and exhaled normally is called 
    the tidal volume. It takes approximately 5 seconds to complete a 
    single cycle. With this information, we can develop an equation for 
    lung capacity as a function of time for normal (resting) breathing. 
    Since sy describes the midpoint of the sine wave, we take the 
    functional residual capacity plus 1/2 the amplitude to give us 
    sy=2200+250=2450 mL. Our period is the time for a single cycle, 
    or p=5 seconds. The volume of air inhaled gives the amplitude, or 
    A=1000 mL. We therefore have the following equation for normal 

    breathing:

    BH

    Unit Summary

    1. The addition and subtraction formulae:

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  • UNIT 2:Sequences

    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,…nth generation.

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    2.1. Arithmetic and harmonic sequences

    2.1.1. Definition

    Activity 2.1

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    Solution

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    2.1.4. Sum of arithmetic sequence

    Activity 2.4

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    2.1.5. Harmonic sequences

    Activity 2.5

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    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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    2.2. Geometric sequences

    2.2.1. Definition

    Activity 2.6

    Take a piece of paper which is in 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 cut, 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 it into two equal parts.
    5. Observe the sequence of numbers you obtained and give 
    the relationship between any two consecutive numbers.
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    2.2.2. General term of a geometric sequence
    Activity 2.7
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    Example 2.28
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    2.2.4. The sum of n terms of a geometric sequence

    Activity 2.9

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    2.3. Convergent or divergent sequences

    Activity 2.11

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    2.4. Applications of sequences in real life

    Activity 2.12

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    20. The third, fifth and seventeenth terms of an arithmetic 
    progression are in geometric progression. Find the common 
    ratio of the geometric progression.
    21. A mathematical child negotiates a new pocket money deal 
    with her unsuspecting father in which he/she receives 
    1 pound on the first day of the month, 2 pounds on the 
    second day, 4 pounds on the third day, 8 pounds on the 
    fourth day, 16 pounds on the fifth day, … until the end of 
    the month. How much would the child receive during the 
    course of a month of 30 days? (Give your answer to the 
    nearest million pounds.)
    22. Find the common ratio of a geometric progression that has 
    a first term of 5 and sum to infinity of 15.
    23. The sum of the first two terms of a geometric progression 
    is 9 and the sum to infinity is 25. If the common ratio is 
    positive, find the common ratio and the first term.
    24. A culture of bacteria doubles every 2 hours. If there are 
    500 bacteria at the beginning, how many bacteria will 
    there be after 24 hours?
    25. You complain that the hot tub in your hotel suite is not 
    hot enough. The hotel tells you that they will increase the 
    temperature by 10% each hour. If the current temperature 
    of the hot tub is 75º F, what will be the temperature of the 
    hot tub after 3 hours, to the nearest tenth of a degree?
    26. The sum of the interior angles of a triangle is 180º, of a 
    quadrilateral is 360º and of a pentagon is 540º. Assuming 
    this pattern continues, find the sum of the interior angles 
    of a dodecagon (12 sides).




  • UNIT 1:Logarithmic and Exponential Equations

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    Objectives
    By the end of this unit, a student will be able to:
    ᇢ solve exponential equations.
    ᇢ solve logarithmic equations. 
    ᇢ apply exponential and logarithmic equations in 

    real life problems.

    3.1. Introduction to Exponential and logarithmic functions

    Activity 3.1

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    Activity 3.2

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    Solution 

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    Activity 3.3

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    3.3. Applications

    Exponential growth

    Activity 3.4

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    Example 3.15

    Jack 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 Frw 3.32 million?

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    Application activity 3.4

    1. A certain cell culture grows at a rate proportional to the 
    number of cells present. If the culture contains 500 cells 
    initially 800 after 24h , how many cells will be there after 
    a further 12h ?
    2. A radioactive substance decays at a rate proportional to 
    the amount present. If 30% of such a substance decays in 
    15 years, what is the half-life of the substance? 
    3. A colony of bacteria is grown under ideal conditions in 
    laboratory so that the population increases exponentially 
    with time. At the end of three hours there are 10,000 
    bacteria. At the end of 5 hours there are 40,000. How many 
    bacteria were present initially? 
    4. A radioactive material has a half-life of 1,200 years. 
    a) What percentage of the original radioactivity of a 
    sample is left 10 years?
    b) How many years are required to reduce the 
    radioactivity by 10%? 
    5. Scientists who do carbon-14 dating use a figure of 5700 
    years for its half-life. Find the age of a sample in which 
    10% of the radioactive nuclei originally present have 
    decayed. 
    6. How much money needs to be invested today at a nominal 
    rate of 4% compounded continuously, in order that it 
    should grow to Frw 10,000 in 7 days? 
    7. The number of people cured is proportional to the number 
    y that is infected with the disease. 
    a) Suppose that in the course of any given year the 
    number of cases of disease is reduced by 20 %. If 
    there are 10,000 cases today, how many years will it 
    take to reduce the number to 1000? 
    b) Suppose that in any given year the number of cases 
    can be reduced by 25% instead of 20% . 
    (i) How long will it take to reduce the number of 
    cases to 1000?
    (ii) How long will it take to eradicate the disease, 
    that is, to reduce the number of cases to less 

    than 1?

    Unit Summary

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  • UNIT4:Trigonometric Functions and their Inverses

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    Objectives:

    By the end of this unit, a student will be able to:
    ᇢ find the domain and range of trigonometric 
    functions and their inverses.
    ᇢ study the parity of trigonometric functions.
    ᇢ study the periodicity of trigonometric functions.
    ᇢ evaluate limits of trigonometric functions. 
    ᇢ differentiate trigonometric functions and their 

    inverses.

    4.1. Generalities on trigonometric functions and 

    their inverses

    4.1.1. Domain and range of six trigonometric 

    functions

    Activity 4.1

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    4.1.2. Domain and range of inverses of 

    trigonometric functions

    Activity 4.2

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    Remark 

    The inverses of the trigonometric functions are not functions, 
    they are relations. The reason they are not functions is that for a 
    given value of x , there is an infinite number of angles at which the 
    trigonometric functions take on the value of x . Thus, the range of 
    the inverses of the trigonometric functions must be restricted to 
    make them functions. Without these restricted ranges, they are 

    known as the inverse trigonometric relations.

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    Example 4.2

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    tan y x = are equivalent

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    Consider the following case:

    If a function repeats every 2 units, then it will also repeat every 
    6 units. So if we have one function with fundamental period 2, 
    and another function with fundamental period 8, we have got no 
    problem because 8 is a multiple of 2, and both functions will cycle 
    every 8 units.
    So, if we can patch up the periods to be the same,we know that 
    if we combine them, we will get a function with the patched up 
    period.
    What we have to do is to find the Lowest Common Multiple (LCM) 
    of two periods.
    What about if one function has period 4 and another has period 5? 
    We can see that in 20 units, both will cycle, so they are fine.
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    4.2. Limits of trigonometric functions and their 
    inverses
    4.2.1. Limits of trigonometric functions
    Activity 4.6
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    We do this until the indeterminate case is removed. Other methods 

    used to remove indeterminate cases are also applied.


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    4.3. Differentiation of trigonometric functions 

    and their inverses

    4.3.1. Derivative of sine and cosine 

    Activity 4.8

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    4.4. Applications

    Simple harmonic motion

    Activity 4.15

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    Solution

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  • UNIT5:Vector Space of Real Numbers

    Introductory activity
    A vector space (also called a linear space) is a collection of 
    objects called vectors, which may be added together and 
    multiplied by numbers, called scalars in this context.
    To put it really simple, vectors are basically all about directions 
    and magnitudes. These are critical in basically all situations.
    In physics, vectors are often used to describe forces, and forces 
    add as vectors do.
    a) Discuss the properties of addition of vectors.
    b) What happens when a vector is multiplied by a 
    scalar(real number)?

    c) Give at least 3 examples of vectors in real life. 

    Objectives
    By the end of this unit, a student will be able to:
    ᇢ define and apply different operations on vectors.
    ᇢ Calculate the scalar and vector product of two 
    vectors.
    ᇢ calculate the angle between two vectors.
    ᇢ apply and transfer the skills of vectors to other 

    area of knowledge

    5.1. Vectors and operations in R3

    Activity 5.1

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    5.3. Magnitude (or norm or length) of a vector

    Activity 5.3

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  • UNIT 6:Matrices and Determinant of Order 3

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    6.1. Square matrices of order three

    6.1.1. Definitions

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    6.1.3. Operations on matrices

    Activity 6.2

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    Multiplying matrices

    Activity 6.4

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    Solution

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    Notice

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    Properties of a determinant

    Activity 6.7

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    6.3. Application

    6.3.1. System of 3 linear equations

    Activity 6.9

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    Let use another useful method.

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    Solution: 

    Let C and M , B, represent the number of serving of chicken, baked 

    potatoes, and milk respectively.

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  • UNIT 7:Bivariate Statistics

    Introductory activity
    In Kabeza village, after her 9 observations about farming, 
    UMULISA saw that in every house observed, where there are 
    a number x of cows there are also y domestic ducks, and then 
    she got the following results of (x,y) pairs: (1,4), (2,8), (3,2), 
    (4,12), (5,10), (6,14), (7,16), (8,6), (9,18)
    a) Represent this information graphically in a 
    ( x y coordinates , ) − .
    b) Chose two points, find the equation of a line joining 
    them and draw it in the same graph. How are the positions 
    of remaining points vis-a -vis this line? 
    c) According to your observation from (a), explain in 
    your own words if there is any relationship between the 
    variation of the number x of cows and the number y of 

    domestic ducks. 

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    Until now, we know how to determine the measures of central 
    tendency in one variable. In this unit, we will use those measures 
    in two quantitative variables known as double series. In statistics, 
    double series includes technique of analyzing data in two variables, 
    when we focus on the relationship between a dependent variable-y 
    and an independent variable-x. The linear regression method will 
    be used in this unit. The estimation target is a function of the 
    independent variable called the regression function which will be 

    a function of a straight line.

    Objectives
    By the end of this unit, a student will be able to:
    ᇢ find measures of variability in two quantitative 
    variables.
    ᇢ draw the scatter diagram of given statistical 
    series in two quantitative variables.
    ᇢ determine the linear regression line of a given 
    series,
    ᇢ calculate a linear coefficient of correlation of a 

    given double series and interpret it

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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), which is a 
    measure of how these two variables change together. 
    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, meaning 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.

    Covariance of variables x and y, where the summation of frequencies

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    Method of ranking
    Ranking can be done in ascending or descending order.
    Example 7.6
    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, 7 such that the smallest value of x 

    will be ranked 1.

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    a) It is required to estimate the value of X for Nepal, where the 
    value of Y is 450.
    i) Find the equation for a suitable line of regression. 
    Simplify your answer as far as possible, giving the 
    constants correct to three significant figures 
    ii) Use your equation to obtain the required estimate 
    b) Use your equation to estimate the value of x for North Korea, 
    where the value of Y was 858. Comment on your answer.
    Solution 
    a) i) Neither variable has been controlled in the given data and 
    since you are required to estimate the life expectancy, X 
    years, when the GDP per head, Y dollars is 160 dollars, it is 
    sensible to use the regression line of X on Y

    The regression line of X on Y has equation

    Unit Summary


    End of Unit Assessment

    1. For each set of data, find:

    a) equation of the regression line of y on x.

    b) equation of the regression line of x on y


    r = 0.95
    Find both equations of the regression lines. Also estimate the value of

    y for x = 30 .

    5. The following results were obtained from records of age (x) and 

    systolic blood pressure Yes of a group of 10 men:

    a) Calculate the Spearman’s coefficient of rank correlation 
    between position in the league and average attendance.
    b) Comment on your results
    21.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). 

    The price is also recorded:

    23.To test the effect of a new drug twelve patients were examined 
    before the drug was administered and given an initial score (I) 
    depending on the severity of various symptoms. After taking the 
    drug they were examined again and given a final score (F). A 
    decrease in score represented an improvement. The scores for 
    the twelve patients are given in the table below:




  • UNIT8:Conditional Probability and Bayes Theorem

    Introductory activity

    A box contains 3 red pens and 4 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. 
    Give more other examples of real life problems involving 
    probability.
    Some academic fields based on the probability theory are statistics, 
    communication theory, computer performance evaluation, signal 
    and image processing, game theory...
    In medical decision-making, clinical estimate of probability 
    strongly affects the physician’s belief as to whether or not a patient 
    has a disease, and this belief, in turn, determines actions: to rule 
    out, to treat, or to do more tests, doctors may use conditional 
    probability to calculate the probability that a particular patient 
    has a disease, given the presence of a particular set of symptoms....
    Some applications of the probability theory are character 
    recognition, speech recognition, opinion survey, missile control 
    and seismic analysis, etc.
    In addition, the game of chance formed the foundations of 

    probability theory

    Objectives

    By the end of this unit, a student will be able to:
    ᇢ use tree diagram to find probability of events.
    ᇢ find probability of independent events.
    ᇢ find probability of one event given that the other 
    event has occurred.

    ᇢ use and apply Bayes theorem.

    8.1. Independent events

    Activity 8.1

    A bag contains books of two different subjects. One book is 
    selected from the bag and is replaced. A book is also selected 
    in the bag. Is the occurrence of the event for the second 

    selection affected by the event for the first selection? Explain



    Example 8.2
    A factory runs two machines. The first machine operates for 80% 
    of the time while the second machine operates for 60% of the time 
    and at least one machine operates for 92% of the time. Do these 

    two machines operate independently?

    Application activity 8.1
    1. A dresser drawer contains one pair of socks with each of 
    the following colors: blue, brown, red, white and black. 
    Each pair is folded together in a matching set. 
    You reach into the sock drawer and choose a pair of socks 
    without looking. You replace this pair and then choose 
    another pair of socks. What is the probability that you 
    will choose the red pair of socks both times?
    2. A coin is tossed and a single 6-sided die is rolled. Find 
    the probability of landing on the head side of the coin and 

    rolling a 3 on the die.

    8.2. Conditional probability

    Activity 8.2
    A bag contains books of two different subjects. One book is 
    selected from the bag and is not replaced. Another book is 
    selected in the bag. Is the occurrence of the event for the second 
    selection affected by the event for the first selection? Explain.
    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 sample space is changed, the events are said to be 
    dependent.
    The probability of an event B given that event A has occurred 

    is called the conditional probability of B given A and is written 

    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.

    Entries in the total row and total column are called marginal 
    frequencies or the marginal distribution. Entries in the body of 
    the table are called joint frequencies. 
    Example 8.6
    Suppose a study of speeding violations and drivers who use car 

    phones produced the following fictional data:

    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).

    Application activity 8.2

    1. A coin is tossed twice in succession. Let A be the event 
    that the first toss is heads and let B be the event that the 
    second toss is heads. Find P(B\A)
    2. Calculate the probability of a 6 being rolled by a die if it is 
    already known that the result is even.
    3. 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?

    8.3. Bayes theorem and its applications









    Thus, the most likely diagnosis in the presence of rebound 
    tenderness/ painfulness is acute appendicitis. Since surgery 
    is generally not required for either pancreatitis or NSAP, but is 
    indicated for appendicitis, it is very useful to know the probability 
    that the patient showing rebound tenderness has appendicitis as 
    compared to the probability that s/he has one of the other two 

    conditions.


    3. The probability of having an accident in a factory that 
    triggers an alarm is 0.1. The probability of it 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?


    End of 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. 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?
    3. A car dealership is giving away a trip to Rome to one of 
    their 120 best customers. In this group, 65 are women, 
    80 are married and 45 married women. If the winner is 
    married, what is the probability that it is a woman?
    4. A card is chosen at random from a deck of 52 cards. It is 
    then replaced and a second card is chosen. What is the 
    probability of choosing a jack and then an eight?
    5. A jar contains 3 red, 5 green, 2 blue and 6 yellow marbles. 
    A marble is chosen at random from the jar. After replacing 
    it, a second marble is chosen. What is the probability of 
    choosing a green and then a yellow marble?
    6. A school survey found that 9 out of 10 students like pizza. 
    If three students are chosen at random with replacement, 
    what is the probability that all three students like pizza?
    7. A nationwide survey found that 72% of people in the 
    United States like pizza. If 3 people are selected at random, 
    what is the probability that all three like pizza?
    8. 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? 
    9. For married couples living in a certain suburb, the probability 
    that the husband will vote on a bond referendum is 0.21, 
    the probability that his wife will vote in the referendum is 
    0.28, and the probability that both the husband and wife 
    will vote is 0.15. What is the probability that:
    a) at least one member of a married couple will vote?
    b) a wife will vote, given that her husband will vote?
    c) a husband will vote, given that his wife does not 
    vote?
    10.In 1970, 11% of Americans completed four years of college, 
    43% of them were women. In 1990, 22% of Americans 
    completed four years of college; 53% of them were women. 
    (Time, Jan.19, 1996).
    a) Given that a person completed four years of college 
    in 1970, what is the probability that the person was 
    a woman?
    b) What is the probability that a woman would finish 
    four years of college in 1990?
    c) What is the probability that in 1990 a man would 
    not finish college?
    11.If the probability is 0.1 that a person will make a mistake 
    on his or her state income tax return, find the probability 
    that:
    a) four totally unrelated persons each make a mistake
    b) Mr. Jones and Ms. Clark both make a mistake and 

    Mr. Roberts and Ms. Williams do not make a mistake.

    12.The probability that a patient recovers from a delicate 
    heart operation is 0.8. What is the probability that:
    a) exactly 2 of the next 3 patients who have this 
    operation will survive?
    b) all of the next 3 patients who have this operation 
    survive?
    13.In a certain federal prison, it is known that 2/3 of the 
    inmates are under 25 years of age. It is also known that 
    3/5 of the inmates are male and that 5/8 of the inmates are 
    female or 25 years of age or older. What is the probability 
    that a prisoner selected at random from this prison is 
    female and at least 25 years old?
    14.A certain federal agency employs three consulting firms 
    (A, B and C) with probabilities 0.4, 0.35, 0.25, respectively. 
    From past experience it is known that the probabilities 
    of cost overrun for the firms are 0.05, 0.03, and 0.15 
    respectively. Suppose a cost overrun is experienced by the 
    agency.
    a) What is the probability that the consulting firm 
    involved is company C?
    b) What is the probability that it is company A?
    15.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?
    References
    [1] A. J. Sadler, D. W. S. Thorning: Understanding Pure 
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    [2] Arthur Adam Freddy Goossens: Francis Lousberg. 
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    Good%20Applied%20calculus.pd