Probability
1.
2.
Concepts
o
dependent and independent events;
o
the product rule for independent events:
o
the sum rule for mutually exclusive events A and B:P(A or B) = P(A) + P(B)
o
the identity: P(A or B) = P(A) + P(B) – P(A and B)
o
the complementary rule:
P(A and B) = P(A) × P(B).
P( not A) = 1 – P(A)
Probability problems using Venn diagrams, Tree diagrams, two-way contingency tables and
other techniques to solve probability problems (where events are not necessarily
independent).
3.
Apply the fundamental counting principle to solve probability problems.
GRADE 12
Counting Methods
Fundamental Counting Principle;
If there are m choices for the first task and n choices for the second task, then there are m×n choices
for the first task followed by the second task.
EXAMPLE 1 : Determine the number of ways a netball team consisting of 7 players can arrange
them selves in a line for a group picture.
We can generalize this result to the case where there are n distinct players.
When we arrange n distinct players in a line, the number of different positions we can find is
n  n  1  n  2   2 1
An arrangement in order of a set of distinct objects, is called a permutation of the set. Learners do
not need to know the word “permutation”
Note that the curriculum does not prescribe the study of permutations and combinations. These are
topics in the mathematical field of combinatorics. However most of the arrangements problems we
teach in the curriculum are in fact permutation problems.
Factorial Notation
The expression n  n  1  n  2   2 1 is called n factorial and written as n!
EXAMPLE 2:Now let us suppose that there are only 4 positions for the picture of the netball team.
Determine the number of different ways the players can be arranged in line for the picture.
The number of different permutations of r objects which can be made from n distinct objects
1
is
n!
n
and usually written as Pr or n Pr
n  r !
EXAMPLE 3: 8 swimmers are hoping to take part in a race, but the pool has only 6 lanes. In how
many ways can 6 of the 8 swimmers be assigned to the lanes?
Exercise 1
1.1
Suppose you are at Telley’s restaurant and like the following items on the
menu:
Dinners
Desserts
Beverages
Big Boy Combination
R18,50
Fudge Sundae
R6,40
Coffee
R1,20
Fried Chicken
R12,40
Apple pie
R4,20
Coca-Cola
R1,20
Shrimp
R17,60
Banana Cream
R5,30
Ginger
R1,80
Beer
Fish and Chips
R8,20
Strawberry Pie
R7,20
Milk
R1,00
Telley’s special steak
R18,20
Cheese Cake
R7,50
Tea
R1,20
Ice Cream
R4,20
(a) If you order a dinner and a beverage, how many different choices have you got?
(b) If you also order a dessert, how many combinations are possible?
(c) You don’t eat shrimps and you are definitely going to have cheese cake, how many
choices of dinner, dessert and beverages do you have?
1.2 How many security codes consisting of three digits can be used from the digits
1; 2; 3; 4; 5; 6; 7; 8; 9 if
(a) Repetition is not allowed
(b) Repetition is allowed
1.3 How many three-character codes can be formed if the first character must be a
letter and the second two characters must be digits?
1.4 Determine how many 4-digit numbers can be formed from the 10 digits
0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 if
(a) Repetitions are allowed.
(b) Repetitions are not allowed.
(c) The last digit must be 0 and repetitions are not allowed.
2
1.5
There are six desks in the front row of a classroom.
(a) In how many different ways can the first 6 learners on the class list be
seated in the front row?
(b) In how many different ways can 6 learners out of a class of 20 be
seated in the front row?
1.6 Four different Mathematics books, six different Accounting books and two
different Geography books are to be arranged on a shelf. How many different
arrangements are possible if
(a) the books in each particular subject must all stand together?
(b)
only Mathematics books must stand together?
1.7 Four (4) different Latin books, 5 different English books and 3 different French books
are arranged in a shelf. How many ways can you arrange them if:
(a) They can be in any order?
(b) Latin books, English books and French books must be together?
(c) Only Latin books must be together?
Counting methods when identical letters are involved
EXAMPLE 4: Write down all possible four letter arrangements that may be formed using the
following
words: (a) DEED
(b) EZEE
(c) ANNA
(d) BULL
(e) MIST
The number of distinct permutations of n objects, of which p are identical to each other, and then q of
the remainder are identical, and r of the remainder are identical, and so on is
n!
where
p ! q ! r !
p  q  r   n
EXAMPLE 5: Copnsider the word. MISSISSIPPI
a) Find the number of distinct random arrangements of the words above.
(b) Determine the probability that each of the words above will start and end with a vowel.
(c) Determine the probability that each of the words above will start and end with the same letter.
Exercise 2
2.1 What is the probability that a random arrangement of the letters BAFANA starts and ends with
an ‘A’?
2.2 If you arrange the letters of the word S O C I O L O G Y what is the probability that
(a) The word will start with an O and end with an O?
3
(b) The three O’s will be together?
2.3
Consider the word LETTER.
(a) How many different letter arrangements can be made out of this word?
(b) What is the probability that such a word will start with the letter E?
(c) What is the probability that such a word will start with the letter E and end with an E?
(d) What is the probability that such a word will start and end with the same letter?
2.4
There are 3 different blue books and 2 different red books on the bookshelf.
(a) In how many different ways can the books be arranged?
(b) In how many different ways can the books be arranged, if all the blue
books should be placed next to each other?
(c )Calculate the probability that the books are arranged in such a way that all the blue
books and all the red books should be placed next to each other?
Nov 2009
4.1
In a company there are three vacancies. The company had identified candidates to fill each
post.
POST
CANDIDATES
Clerk
Craig, Luke and Tom
Sales representative
Ann, Sandile, Sizwe and Devon
Sales manager
John and Debby
4.1.1
In how many different ways can these three posts be filled?
4.1.2
If it is certain that Craig will get the job as clerk, in how many
different ways can the three posts be filled?
4.2
(3)
(2)
There are 20 boys and 15 girls in a class. The teacher chooses individual
learners at random to deliver a speech.
4.2.1
Calculate the probability that the first learner chosen is a boy.
4.2.2
Draw a tree diagram to represent the situation if the teacher
(1)
chooses three learners, one after the other. Indicate on your
diagram ALL possible outcomes.
4.2.3
(4)
Calculate the probability that a boy, then a girl and then another
boy is chosen in that order.
(3)
4
4.2.4
Calculate the probability that all three learners chosen are girls.
4.2.5
Calculate the probability that at least one of the learners chosen
(2)
is a boy.
4.3
(3)
In a Mathematics quiz, two teams work independently on a problem.
They are allowed a
maximum of 10 minutes to solve the problem. The probabilities that each team will solve the
problem are
1
1
and
respectively. Calculate the probability that the problem will be solved
2
3
in the ten minutes allowed.
(4)
March 2010
4
P(A) = 0,3 and P(B) = 0,5. Calculate P(A or B) if:
4.1
A and B are mutually exclusive events
(2)
4.2
A and B are independent events
(3)
5.
At a school for boys there are 240 learners in Grade 12. The following
information was gathered about participation in school sport.
o
122 boys play rugby (R)
o
58 boys play basketball (B)
o
96 boys play cricket (C)
o
16 boys play all three sports
o
22 boys play rugby and basketball
o
26 boys play cricket and basketball
o
26 boys do not play any of these sports
Let the number of learners who play rugby and cricket only be x.
5.1
Draw a Venn diagram to represent the above information.
(4)
5.2
Determine the number of boys who play rugby and cricket.
(3)
5.3
Determine the probability that a learner in Grade 12 selected at random:
(Leave your answer correct to THREE decimal places.)
5.3.1 only plays basketball.
(2)
5.3.2 does not play cricket.
(2)
5.3.3 participates in at least two of these sports.
(2)
5
Exercises
1. M1, M 2 and M 3 are 3 machines in a factory that manufactures plastic containers.
They produce 25%, 30% and 45% of the total production of the factory.
Of the products of M1, M 2 and M 3 is 18%, 5% en 2% are defective.
A random sample is drawn of the products
1.1.
Represents this data in a tree diagram.
1.2
Determine the probability that
(6)
3.2.1.
The defective products are manufactures by machine M1
(2)
3.2.2.
The products are defective
(3)
[11]
2.
In a grade 11 class of 30 girls, 15 plays tennis and 20 plays netball. If 8 girls practice
both sporting codes, calculate the probability that a randomly chosen girl:
2.1
only plays netball
2.2
plays at least one sporting code
2.3
practice none of the two sporting codes.
3. Figures obtained from the city’s police department seem to indicate that all of the motor
vehicles stolen, 80% were stolen by syndicates to be sold off and 20% by individual
persons for their own use.
Of those vehicles presumed stolen by syndicates:

24% were recovered within 48 hours

16% were recovered after 48 hours

60% were never recovered
Of those vehicles presumed stolen by individual persons:

38% were recovered within 48 hours

58% were recovered after 48 hours

4% were never recovered
(i)
Draw a tree diagram for the above information
(ii)
Calculate the probability that if a vehicle were stolen in this city, it would be
stolen by a syndicate and recovered within 48 hours
(iii)
Calculate the probability that a vehicle stolen in this city will not be recovered
6
4.
A bag contains 5 red and 7 green marbles. Two marbles are drawn successively with
out replacing. Calculate the probability that both marbles are of the same colour by
sing a tree diagram
5.
(6)
A study on speeding fines and gender yielded the following results (this is a fictitious
scenario).
Speed fine
No speed fine
total
Male drivers
45
25
70
Female drivers
35
45
80
total
80
70
150
5.1
How many persons participated in this study
5.2
Calculate the following probabilities
5.2.1 P(male drivers)
5.2.2 P(speed fines)
5.3
Are the events male driver and speed fine independent. Justify your answer through
calculations.
6.
The data below was obtained from the financial aid office at a certain university.
Receiving
Not receiving
financial aid
financial aid
Undergraduates
4 222
3 898
8 120
Postgraduates
1 879
731
2 610
TOTAL
6 101
4 629
10 730
(i)
(ii)
TOTAL
Determine the probability that a student selected at random is:
(a)
receiving financial aid
(2)
(b)
a postgraduate student and not receiving financial aid
(2)
(c)
an undergraduate student and receiving financial aid
(2)
Are the events of being an undergraduate and receiving
financial aid independent? Show all relevant workings to
support your answer.
7.
(4)
The following contingency table shows the preferred pastimes for boys and girls
respectively:
7
Reading
TV games
Hiking
Total
Boys
16
18
e
45
Girls
a
c
f
55
Total
b
d
g
h
7.1
Complete the missing numbers indicated a h
7.2
Are the events boys and TV games dependent of independent. Justify your answer
7.3
Are the events boys and girls exhaustive
7.4
Are the events boys and girls complementary
8
Scatter plots, Correlation and Regression
Up to this point in the curriculum we have been dealing with univariate data only. We have looked at
both group and ungrouped data. The data we have dealt with could also be continuous or discrete data.
In Grade 12 we will also study bivariate data, and look at how to display such data graphically and
how to determine the relationship between the two variables, using a regression and correlation
concepts.
A Scatter plot is a statistical graph showing the relationship two variables in bivariate data.
Types of scatterplots
The realationship is:
Exponential function
Quadratic (parabola)
Linear (straight line)
The line or curve drawn in the diagrams above is called the line (curve) of best fit
Learners are expected to “suggest intuitively whether a linear, quadratic or exponential function
would best fit the data”. The line of best fit is an estimate.
Regression anmalysis
Regression attempts to show the relationship between the variables by providing a function which
best indicates the trend of the relationship
We can mathematically determine the equation of the regression line (least squares regression line).
The form is: 𝒚 = 𝑨 + 𝑩𝒙, where B is the gradient of the line and A the 𝑦-intercept.
Note that the least squares regression line is always a straight line.
The reason for the equation in the form 𝒚 = 𝑨 + 𝑩𝒙 is that we can extend to Quadratic regression:
with equation : 𝒚 = 𝑨 + 𝑩𝒙 + 𝑪𝒙𝟐 . Similarly we may also get Exponential regression: : 𝒚 = 𝑨 𝒆𝑩𝒙
However we will only focus on linear regression
9
Correlation
Correlation is concerned with whether there is a relationship between the two variables as
shown below. In the diagram the datapoints lie in a narrow band around the straight line
A strong linear correlation between the variables and this enables us to use a straight lne as the line of
best fit. With this knowledge we can make decisions as well as presdictions with respect to the data.
Consider the two examples below and note the use of the tems POSITIVE and NEGATIVE
Strong negative correlation
Strong positive correlation
10
Correlation is a numerical measure of the strenght of the relationships between variables and is
denoted by the letter r, called the coefficient of correlation. The value of r ranges between -1 and 1.
The closer the r is to -1 or 1, the stronger the relationship between variables x and y.
-1
Strong negative correlation
-0,5
0
0,5
1
Strong positive correlation
No correlation
1.
The sign of r indicates the direction of a slope .
2.
The numerical value of r indicates the strength of the relationship, i.e. +0,9 indicates a strong
positive correlation, -0,9 indicates the strong negative correlation.
y
y
11
Exam Type questions
12
13
14
15
QUESTION 5
A learner conducted an experiment to investigate the relationship between age and resting heart rate (in
beats per minute). He sought the assistance of the local clinic. The information for 12 people is shown
in the table below.
Age
Resting heart rate
(beats per minute)
59
32
42
50
22
39
21
20
27
40
29
47
88
74
74
93
85
71
78
82
70
75
95
75
5.1
Represent the data in a scatter plot.
(3)
5.2
Determine the equation of the least squares line.
(4)
5.3
Draw the least squares line on the scatter plot.
(2)
5.4
Calculate the correlation coefficient for the data.
(2)
16
5.5
5.6
Use the correlation coefficient to comment on the relationship between age and the resting
heart rate.
(2)
If a learner uses the least squares line to predict the resting heart rate of a 45-year-old person,
(2)
will his answer be reliable? Motivate your answer.
[15]
QUESTION 1
A recording company investigates the relationship between the number of times a CD is played by a
national radio station and the national sales of the same CD in the following week.
The data below
was collected for a random sample of 10 CDs. The sales figures are rounded to the nearest 50.
Number of times CD is
played
Weekly sales of the CD
47
34
40
34
33
50
28
53
25
46
3 950 2 500 3 700 2 800 2 900 3 750 2 300 4 400 2 200 3 400
1.1
Identify the independent variable.
(1)
1.2
Draw a scatter plot of this data on the grid provided on DIAGRAM SHEET 1.
(3)
1.3
Determine the equation of the least squares regression line.
(4)
1.4
Calculate the correlation coefficient.
(2)
1.5
Predict, correct to the nearest 50, the weekly sales for a CD that was played 45 times by the
1.6
radio station in the previous week.
(2)
Comment on the strength of the relationship between the variables.
(1)
[13]
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