CSEC Mathematics Section 5
Probability
Probability is defined as the measure of how
likely an event is to occur. The probability of an event is a number from 0 (the impossible event) to 1 (the certain event). For example:
The probability of a person having three heads is 0.
The probability of a person walking on the sun is zero.
The probability that a person will die is 1.
The probability of the sun setting in the west is 1.
It can be denoted as .
Exercise
Answer the following.
1. A fair coin is tossed once and the symbol that appears on top is observed. Calculate the
probability that a:
a. head appears b. tail appears
2. A bag contains 60 marbles of which 15 are red and the remainders of marbles are green. a. How many green marbles are in the bag? b. What is the probability of drawing a green marble?
d. What is the chance of drawing a yellow marble?
e. What is the probability of drawing either a red marble or a green marble?
f. If 15 green marbles are removed from the bag, what is the chance now of drawing a green
marble?
3. A card is chosen from a standard pack of 52 playing cards. What is the probability that the card is: Take cards to class
4. A whole number from 1 to 11 is chosen at random. What is the probability of choosing an odd number?
5. A teacher chooses a student at random from a class of 30 girls. What is the probability that the student chosen is a girl?
6. A glass jar contains 5 red, 3 blue and 2 green jelly beans. If a jelly bean is chosen at random from the jar, what is the probability that it is not blue?
7. An ordinary, fair, 6-sided die is rolled. What is the probability of rolling:
a) 4? b) a number less than 3?
8. The scores obtained by students in a test were recorded in a frequency table as follows:
Marks (x) Number of Students (Frequency, f)
0 1
1 4
2 7
3 10
4 18
5 10
What is the probability that a student chosen at random scores:
a. 3 marks? Ans: 10/50 b. less than 2 marks? Ans: 5/50 c. 3 or more marks? Ans: 38/50
below.
Number of Goals
0 1 2 3 4 5
Frequency 4 3 5 5 6 7
What is the probability that the number of goals scored in the 30 matches is:
a. 4 b. less than 3 c. 4 or more
10. The Histogram below shows the number of hours a group watched television during a
particular evening.
Number of T.V. hours 0
5 10 15 20
Calculate the probability that a person selected at random from this group watched television for: a. 5 hours. b. 3 hours. c. 4 or more
hours d. 1 or 2 hours e. less than 2 hours
Statistics
Statistics is concerned with the scientific
methods for collecting, organizing, interpreting, summarizing, presenting and analyzing data. It involves stating a valid conclusion and making reasonable decisions on the basis of such
analysis.
A single value which represents or characterizes a group (or set of data) as a whole is called a
statistical average or a measure of central tendency.
The three statistical averages (or measures of central tendency) are arithmetic mean, median and mode.
The Mean
The Arithmetic mean (or average) of a set of
quantities is the sum of the quantities divided the total number of quantities. That is,
. We note, The sum of the quantities = The Arithmetic mean (or average) the total number of quantities.
That is, total = mean × n. Exercise
Answer the following.
1. Calculate the mean of the set of scores 13, 6, 10, 7, 8. Ans: 8.8
2. Evaluate the mean of the numbers 17, 16, 5, 7, 1, 13, 5 Ans: 9.14
a. man of height 169 cm leaves the choir.
Ans: 155.5 cm
b. woman of height 165 cm joins the original choir. Ans: 157.8 cm
4. John received the following scores (in %) on five chemistry tests, 72, 86, 92, 63 and 77.
a. What is John’s average (mean) score on the chemistry tests? Ans: 78%
b. What score must John earn on his sixth test so that his average for all six tests will be 80?
Ans: 90%
5. The mean mass of three children is 38
have the same mass. Calculate the mass of one of the twins. Ans: 34kg
6. Betty’s average marks for eight examination papers was 74.5. How many marks did she score altogether? Ans: 596
7. Adam bought five books at $9.48 each and three books at $5.32 each. What was the mean amount that Adam paid for a book?
Answer $7.92
8. The average mark of 25 students in a
Mathematics test was 48. Calculate what the
average mark would have been, if a student who scored 84 marks had been absent.
Answer 46.5
140 cm, 150cm, calculate the height of the fifth student. Answer 165cm
10. In nine completed innings, a batsman’s
average score was 47. After a further innings, his average score increased to 51.
a. What was his total runs after nine completed innings? Answer 423 runs
b. Calculate his total runs scored up to the 10th
innings. Answer 510 runs
c. Hence, determine the total runs scored in the 10th innings. Ans: 87 runs
11. Calculate the average age of five girls, given that three of them are each 14 years 6 months of age and the other two are each 16 years 2 months of age.
The Mean from an Ungrouped Frequency Table
An ungrouped data does not have a class interval.
Exercise
Answer the following
1. The number of goals scored by a team in 30 matches was recorded in the frequency table
below. Calculate the mean number of goals scored by the team over the series.
Number of Goals
0 1 2 3 4 5
Frequency 4 3 5 5 6 7
Ans: Mean = 2.9 ≈ 3 goals
Marks (x) Number of Students (Frequency, f)
0 1
1 4
2 7
3 10
4 18
5 10
Calculate the mean mark of the scores in the test.
Ans: Mean = 3.4
The Mean from Grouped Frequency Table
A grouped data consist of class intervals.
We calculate the mean of a grouped data by first finding the mid-point of each class, then
The lower limit for a class is the value on the left of the class interval. The upper class limit is the value on the right of the class interval.
The mid-point of a class interval =
.
Exercise
Answer the following
1. The table below shows the masses of peas in grams.
Mass in Grams Number of Peas
3 – 7 3
8 – 12 8
13 – 17 12
18 – 22 10
Calculate the mean mass of the peas.
Ans: Mean = 16.25 g
2. The frequency distribution of the marks
awarded to 100 candidates in an examination is shown in the table below. Evaluate the mean
mark to the nearest whole number.
Marks Number of Candidates
1 – 10 13
11 – 20 23
21 – 30 36
31 – 40 20
41 – 50 8
Ans: 24 marks
The median of a set of values/data/scores is the middle or central value of the scores which are arranged in ascending (smallest to largest) or descending (largest to smallest) numerical order. If there are two middle values, then the median is the mean of the two middle values.
Exercise
Answer the following.
1. Determine the median of the following set of scores:
a. 8, 4, 5, 0, 9, 3, 8 b. 16, 10, 23, 25, 10, 19 2. The scores obtained by students in a test
were recorded in a frequency table below. Find the median mark.
(Frequency, f)
0 1
1 2
2 4
3 3
3. The table below shows the masses of peas in grams. Determine the median mass.
Mass in Grams Number of Peas
3 5
4 3
5 1
6 2
7 4
The Mode
The mode is the value or observation which
we say that the distribution or data is bimodal. If three modes exist, we say that the distribution or data is trimodal.
Exercise
Answer the following
1. Determine the mode for the following data: a. 12, 8, 14, 12, 8, 6, 14, 8
b. 10, 2, 3, 8, 7, 2, 10, 5
2. The table below shows the masses of peas in grams. Determine the modal mass.
Mass in Grams Number of Peas
3 5
4 3
5 1
6 2
7 4
below. Determine the modal number of goals scored by the team over the series.
Number of Goals
0 1 2 3 4 5
Frequency 4 3 7 5 6 2
4. The frequency distribution of the marks
awarded to 100 candidates in an examination is shown in the table below. Evaluate the modal class interval.
Marks Number of Candidates
1 – 10 13
11 – 20 23
21 – 30 36
31 – 40 20
41 – 50 8
Variables can be divided into two main types – qualitative and quantitative.
A qualitative variable is defined as a variable which describes a characteristic. For example, ‘the height of a person can be described as short, average or tall.
A quantitative variable is defined as a variable which can be given a numerical value.
Quantitative variables are said to be of two distinct types – discrete and continuous.
A discrete variable is defined as a variable which can only take certain definite values, usually
whole numbers. For example, the number of
number of mangoes on the tree cannot be 24.635.
A continuous variable is a variable which can take any value within a given range and can be obtained by measurement. For example, ‘a
person’s height, which may vary from birth to adulthood’.
Exercise
Answer the following.
State whether the following variables are discrete or continuous.
1. The weight of a baby during the first year of its life on earth.
2. The number of students in room 4.
4. The number of cars at Old Harbour Campus.
Ungrouped and Grouped Data
A grouped data is one in which each observation is placed in its respective class interval. An
ungrouped data does not have a class interval.
Frequency Table with Ungrouped Data
The frequency of an event (or observation or
score) is defined as the number of times an event has occurred. A frequency table is used to
summarize a set of data. It records the frequency or number of times each value in the table
the data, the second is the tally chart and the third shows the frequency of each data.
Exercise
Answer the following.
1. A class of 30 students counted the number of books in their bags on a certain day. The
number of books in each bag is shown below.
MAY 2014 – Question 7
5 4 6 3 2 1 7 4 5 3 6 5 4 3 7 6 2 5 4 5 5 7 5 4 3 2 1 6 3 4
a. Copy and complete the frequency table for the data shown above.
Number of Books(x)
Tally Frequency (f)
x f
1 ll 2 2
2 lll 3 6
4 5 6 7
b. State the modal number of books in the bags for the sample of students.
c. Using the table in (a) above, or otherwise, calculate the:
i. total number of books Ans: 127
ii. mean number of books per bag. Ans: 4.23 ≈ 4 books
d. Determine the probability that a student chosen at random has less than 4 books in his/her bag.
5, 7, 3, 0, 1, 4, 5, 2, 1, 3, 4, 2, 1, 5, 2, 5, 2, 0, 3, 1, 4, 0, 3, 2, 4, 2.
a. Draw a frequency table for the given data. b. How many persons participated in the dart competition?
c. What is the modal score?
d. How many participants scored: i) more than 2?
ii) between 0 and 4? iii) at most 2?
iv) at least 4?
i) the maximum score? Ans: 3.85%
ii) 3 or 4? Ans: 30.77%
Frequency Table with Grouped Data
Sometimes the data under consideration has such a large range of values that it is most useful to collect these values into groups (or classes). The class interval is defined as the size of the group (or class) chosen.
Exercise
Answer the following.
Copy and complete the grouped frequency table to represent the data given.
7 10 6 4 11 6 1 5 2 8 6 1 2 5 10 13 10 11 4 1 2 3 1 2
1 3 4 5 6 10 11 15 14 2 6 1 2 12 3 10 5 6 7 11 14 8 9 5
Number of Pens (x)
Tally Frequency
1 – 5 6 – 10 11 – 15 16 – 20
a. What is the modal class?
Some Scales on a Graph
1 cm : 1 unit 1 cm : 2 units 1cm : 5 units 1 cm : 10 units
2 cm : 1 unit
2 cm : 100 units 2 cm : 500 units 2 cm : 1000 units
Line Graph
Line graphs are useful means of displaying
statistical data to examine trends or growth over a period of time. It is used in business where
companies may predict growth in income or sales over a period of time. In medicine,
A line graph is constructed by joining a set of points together in a consecutive manner.
Exercise
Answer the following.
1. The line graph below shows the monthly
sales, in thousands of dollars, at a car dealership for the period July to November 2014.
MAY 2015 – Question 7
a) Complete the table below to show the sales for each month.
Month July August September October November Sales in $
Thousands
13 36
b) i. Between which two consecutive months was there the greatest increase in sales?
iii. What feature of the line graph enables you to infer that the increase in sales between two consecutive months was the greatest or the
smallest?
c. Calculate the mean monthly sales for the period July to November 2014. Ans: $21,800
d. The total sales for the period July to December was $130,000.
i. Calculate the sales, in dollars, for the month of December. Ans: $21,000
2. A businessman recorded his profits over a five year period in the table below.
Year 2006 2007 2008 2009 2010 2011
Profit 10000 12000 13000 10000 15000 18000
a. Represent this data on a line graph.
b. Comment on any noticeable trend on your graph.
c. Which period had the largest increase in profit?
d. Which period had the lowest increase in profit?
e. What was the smallest profit observed over the five year period?
f. What do you expect to happen to the profit in the year 2012? Why?
3. SEE JANUARY 2011 – QUESTION 6
4. The table below shows the quantity of
bananas (in tonnes) grown annually on a farm over the period 1989 to 1993.
Year Quantity of bananas in tonnes
1989 250 1990 450 1991 200 1992 500 1993 600
a. Draw a line graph to represent this data.
ii. In which year was the quantity of bananas the lowest?
iii. During which period was there a decrease in the quantity of bananas?
iv. How many bananas were produced in the year 1990?
v. During which period was there the largest increase in the quantity of bananas?
Bar Charts
the size of the data it is representing. Bar charts are mostly used to represent discrete data.
Exercise
Answer the following.
1. JANUARY 2008 – QUESTION 5
In a survey, all the boys in a Book Club were
asked how many books they each read during the Easter Holiday. The results are shown in a
histogram below.
Number of T.V. hours
0 15
0 1 2 3 4 Number of Boys
a. Draw a frequency table to represent the data shown in the graph.
b. How many boys are in the Book Club? c. What is the modal number of books read? d. How many books did the boys read during the Easter Holiday?
e. Calculate the mean number of books read. f. What is the probability that a boy chosen at random read three or more books?
3. The table below shows the quantity of
bananas (in tonnes) grown annually on a farm over the period 1989 to 1993. Draw a bar graph to represent the information given.
Pie Chart Types of cars Frequency Honda 55 Toyota 70 Benz 50 BMW 35 Fiat 20
Year Quantity of
bananas in tonnes 1989 250 1990 450 1991 200 1992 500
a. Draw a bar chart to represent the information given.
b. How many bananas were grown over the period 1989 to 1993?
a. Draw a bar chart to represent the information given.
A pie chart is a circular diagram, which is
another way of representing statistical data. The circle is divided into sector of varying sector
angles or areas. Each sector angle or area is directly proportional to the size of the
information it is representing.
Exercise
Answer the following.
1. The sum of $180 was shared among four friends as shown in the pie chart below.
a) Determine the amount of money received by:
i. Kaias ii. Tony iii. James iv. Hugo b) What is the sector angle for:
i. Kaias ii. Tony iii. James iv. Hugo
2. Ted spent $60 on food, $40 on transportation and $80 on phonecards.
a. How much money did he spend altogether? b. Determine the sector angle that will represent the amount spent on each quantity.
c. Construct a pie chart of radius of 5 cm, to represent the information given.
3. An agricultural plot of 24 hectares is divided for planting 6 hectares of potatoes, 4 hectares of peas, 10 hectares of corn and the remainder is planted in carrots.
b. Determine the sector angle that will represent the area to be planted by each crop.
c. Construct a pie chart of radius of 5 cm, to represent the information given.
4. The table below shows the number of
Graduates by subject from a Teacher’s Training College in 1992.
Subject Mathematics English Physics Chemistry History No. of
Teachers
17 25 10 15 23
a. Calculate the total number of teachers that graduated from the training college in 1992.
b. Determine the sector angle that will represent the number of teachers graduating in each
c. Hence, construct a pie chart of radius 5 cm to represent the information given above.
Grouped Frequency Distribution – Class Intervals, Class Limits, Class Boundaries, Class Mid-points, Class Size
Consider: The masses of 60 students correct to the nearest kilograms as shown the frequency table below.
Mass (kg) Frequency
35 – 39 14
40 – 44 12
45 – 49 13
50 – 54 11
55 – 59 10
NB A grouped frequency table consist of class intervals. E.g. 35 – 39, 45 – 49, 55 – 59 and so. Class Interval
The third class interval is 45 – 49
Class Limits
The class limits are the end values of a class interval. Each class interval has two limits,
namely, the lower class limit (the value on the left) and the upper class limit (the value on the right).
The lower class limit for the first class interval is 35 kg
The lower class limit for the third class interval is 45 kg
The upper class limit for the second class is 44 kg
Class Boundaries
Lower Class Boundaries
(L.C.B.)
Class Intervals
Upper Class Boundaries
(U.C.B.)
34.5 35 – 39 39.5
39.5 40 – 44 44.5
44.5 45 – 49 49.5
49.5 50 – 54 54.5
54.5 55 – 59 59.5
Class Mid-point/Class Mark
35 – 39 40 – 44 45 – 49 50 – 54 55 – 59
59.5 Class
Boundaries Class Intervals
The Class Mid-point =
The class mid-point for the 1st class interval =
kg
OR
The Class Mid-point =
The class mid-point for the 1st class interval =
kg
The width of a class interval (or class size) is the difference between the upper and lower class boundaries.
The width of the first class interval is U.C.B – L.C.B. = 39.5 – 34.5 = 5 kg
NOTE: The width of a class is not equal to the upper limit – the lower limit.
Exercise
Answer the following.
1. The scores obtained by 100 children in a competition are summarized in the table below.
JANUARY 2013 – Question 7
point (x) frequency
0 – 9 4.5 8 36
10 – 19 14.5 13 188.5
25
22 20
50 – 59 12
Total 100
a. Copy and complete the table to show the: i) class mid-points
ii) values of “f x”
iii) State the modal class interval.
iv) State the class interval in which a score of 19.4 would lie.
v) State the class interval in which a score of 39.6 would lie.
Ans: 31.4
c. Explain why the value of the mean obtained in (b) is only an estimate of the true value.
d. In order to qualify for the next round of the competition, a student must score at least 40 points. What is the probability that a student selected at random qualifies for the next round?
Ans: P(score at least 40 points) = 0.32
Frequency Polygon
A frequency distribution can be represented graphically by drawing a line graph called a frequency polygon.
interval against the corresponding frequency and then drawing straight lines, in an consecutive order.
Exercise
Answer the following.
1. The heights of a sample of seedlings were measured to the nearest centimetre and then
arranged in class intervals as shown in the
table below. JANUARY 2014 – Question 7
Height (cm) Mid-point Frequency
3 – 7 5 0
8 – 12 10 3
13 – 17 15 12
14
a. Copy and complete the table by inserting the: i. missing values for the class intervals
ii. midpoints of each class interval
b. How many seedlings were in the sample? c. For the class interval written as “8 – 12” in the table above, write down the:
i. lower class limit
ii. upper class boundary iii. class width
frequency polygon to represent the data as shown in your table at (c).
2. The table below shows the scores obtained by some students on a mathematics test.
a. Copy and complete the table below.
Marks Frequency Mid-point (x)
1 – 10 2 11 – 20 7 21 – 30 11
10 14 6
5 students on the y-axis (vertical axis). Draw a frequency polygon to represent the data.
3. The table below shows the length of different leaves on an apple tree.
a. Copy and complete the table below. Lengt
h (x m
m)
Freque
ncy Mid-point (xmm)
11 –
15 2
16 –
20 4
21 –
25 8
b. Use a scale of 2 cm to represent 10 mm on the x-axis and 2 cm to represent 10 leaves on the y-axis. Draw a frequency polygon to represent the data.
Histogram
A histogram consists of bars joined together, where the height of each bar corresponds to the frequency of the data. The histogram for a
grouped data is drawn by plotting either the class boundaries or class mid-points along the x-axis against the corresponding frequencies.
Exercise
1. SEE JANUARY 2012 – QUESTION 7
An Histogram is given to answer questions.
2. The table below shows the scores obtained by some students on a mathematics test.
a. Copy and complete the table below.
Marks Frequency Mid-point (x)
1 – 10 2 11 – 20 7 21 – 30 11
b. Use a scale of 2 cm to represent 2 students on the y-axis (vertical axis). Draw a histogram to represent the data above.
3. The table below shows the length of different leaves on an apple tree.
b. Use a scale of 2 cm to represent 10 mm on the x-axis and 1 cm to represent 2 leaves on the y-axis. Draw a histogram to represent the data above.
The Range (Page 401, Vol. 1, R. Toolsie)
The range of a set of numbers is the difference between the largest and smallest number in the set. For the set of numbers 5, 3, 6, 4, 7, 5, the
range is sometimes given by quoting the smallest and largest numbers or indicated as 3 – 7. The range of this set of numbers is 7 – 3 = 4.
1. The basic wages of workers in a factory are: $175, $160, $195, $149, $185, $167, $148.
Calculate the range of the basic wages.
Ans: $47
2. A comparison was made of the price of a same sized bread at different shops. The prices are $250.35, $249.99, $265.20, $262.50,
$255.00, $264.60. What is the range of the prices for the bread? Ans: $15.21
Interquartile Range and Semi-interquartile Range
one-quarter of data lies. The lower quartile Q1 is the middle value of the bottom half of the data. The second (or middle) quartile Q2 is the value
below which one-half of the data lies. This
quartile is known as the median. The median Q2
is the middle value of the whole data.
The third (or upper) quartile Q3 is the value
below which three-quarters of the data lies. The upper quartile Q3 is the middle value of the top
half of the data.
The semi-interquartile range (S.I.Q.R.) = (Q3 –
Q1)
Exercise (Page 403, Vol. 1 – R. Toolsie)
Find the median, interquartile range and semi-interquartile range for the following heights stated in cm:
a. 153, 168, 164, 151, 166, 169, 165
Ans: Median (Q2) = 165
I.Q.R. = 15 S.I.Q.R. = 7.5
b. 168, 153, 164, 151, 167, 166, 169, 165
Ans: Median (Q2) = 165.5
I.Q.R. = 9 S.I.Q.R. = 4.5
c. 163, 158, 154, 161, 156, 159, 155
Ans: Median (Q2) = 158
S.I.Q.R. = d. 158, 163, 154, 161, 157, 156, 159, 155
Ans: Median (Q2) = 157.5
I.Q.R. = 4.5 S.I.Q.R. = 2.25
Cumulative Frequency Curve
The cumulative frequency is always shown on the y-axis (vertical axis).
There are two ways of drawing a cumulative frequency curve (or ogive).
in a consecutive manner, in order to obtain the ogive.
Another way of drawing the cumulative
frequency curve is by plotting the cumulative frequency (on the y-axis) against the
corresponding upper class boundary (on the
x-axis) for each class interval. A smooth curve is then drawn through the points, from the x-axis, to obtain the ogive. (Focus on the second
method based on the CSEC Mathematics Syllabus).
For a grouped data, the lower quartile (Q1) is the
value corresponding to the nth term, where n is
the total frequency of the data. The median (Q2)
upper quartile (Q3) is the value corresponding to
the nth term.
NOTE: It is very important to use the most appropriate scale on your Cartesian plane. We can use a scale of: 1 cm to 1 unit, 2 cm to 1 unit, 1 cm to 5 units, 1 cm to 10 units or 1 cm to 50 units and so on, for each axis.
Upper class limit Or
Upper class boundary Cumulative
frequency
0
Ogive
Q1 Q2 Q3
y
Exercise
Answer the following.
1. The masses of 60 parcels collected at a post office were grouped and recorded as shown in the histogram below. JANUARY 2015 – 7
a. i) Complete the table below to show the information given in the histogram.
Mass(kg) No. of Parcels Cumulative Frequency
1 – 5 4 4
6 – 10 10 14
11 – 15
26 – 30
b. On graph paper, use a scale of 2 cm to represent 5 kg on the x-axis and 2 cm to
represent 10 parcels on the y-axis, draw the cumulative frequency curve for the data.
c. Use your graph drawn to estimate the median mass of the parcels.
2. The table below shows the scores obtained by some students on a mathematics test.
a. Copy and complete the table below.
Marks(x) Frequency Upper Class Boundary
Cumulative frequency
1 – 10 2
11 – 20 7 21 – 30 11
10 14 6
data.
c. Determine the lower quartile, median and upper quartile from your graph.
d. Use the upper limit to draw a cumulative frequency curve to represent the data.
3. The table below shows the amount, to the
nearest dollar, spent by a group of 40 students at
the school canteen during a period of one
week. MAY 2013 - 7
Amount Spent ($)
Number of Students
Cumulative Frequency
1 – 10 3 3
31 – 40 11
8
2
a. Copy and complete the table to show the cumulative frequency.
b. Using a scale of 1 cm to represent $5 on the horizontal axis and 1 cm to represent 5 students on the vertical axis. Use the upper limit to draw the cumulative frequency graph for the data.
c. Use your graph to estimate the:
ii. probability that a student chosen at random spent less than $23 during the week.
4. The table below shows the length of different leaves on an apple tree.
a. Copy and complete the table below.
Leng th (x m
b. Use a scale of 2 cm to represent 10 mm on the x-axis and 2 cm to represent 10 leaves on the y-axis. Draw a cumulative frequency curve to represent the data.
c. Determine the lower quartile, median and upper quartile from your graph.
