Notesale: Turn your study into money

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CHAPTER 1
Probability
The idea around probability, chance and possibility is quite old and is commonly used in
everyday speech
...

Probability is a concept founded on solid mathematical theory, and can be a powerful aide in
important decision making
...
In these situations accurate probability calculations are of
paramount importance (Swanepoel and Allison, 2011)
...
In some situations only
sample data is available, in such cases probability theory in used to evaluate the conclusions
made about the population
...


1
...

Sample space: The sample space of an experiment is a collection of all possible outcomes and
is denoted by (
)
...
) is a collection of some of the
outcomes
...


See Example A and B (Rice, p
...

We make use of Venn-diagrams to illustrate certain concepts of probability theory
...


Intersection: The intersection of two events A and B is the event that both A and B will take
place, denoted by

...
Therefore all the outcomes in

that are not in A
...

Disjoint: If two events A and B are disjoint (mutually exclusive) it means that if event A
occur event B cannot occur and vice versa, therefore

...
3)

Laws of set theory:
i)

Commutative law:

ii)

Associative law:

iii)

Distributive law:

iv)

and

v)

,

and
and
and

and

vi)

and

vii)

1
...
The following axioms are always satisfied:
i)
ii)

If

iii)

If

then
and

In general, if

are disjoint, then


...


Property D:

,

)

A

B

Example 2
Two fair coins are tossed and the outcomes are recorded:

...

Calculate


...


See Example A (Rice, p
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1
...
If each of the outcomes have the

...
Then


...
6) and Example B (Rice, p
...

In the previous examples it is clear that the number of outcomes must be known to calculate
the probability
...

Sometimes the number of outcome is large and it is difficult to count them
...

Multiplicative principle: If one experiment has m outcomes and another experiment has n
outcomes, then there are a total of mn outcomes for the two experiments
...
Then there
are 5x4 = 20 different choices you can make
...
8)
...


5

Example 5:
A truck driver can take three different routes in order to get from city A to city B, then one of
four routes from city B to city C, and finally one of three routes from city C to city D
...

See Example C (Rice, p
...

1
...
2 Permutations and Combinations
Permutations: Ordered set of elements also called orderings
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elements be chosen from a set C

i)

Sampling with replacement: Each time an element is chosen it is chosen from the
entire set C (an element can be chosen more than once)
...
This is done in
many ways (multiplication principle)
...
Each of the remaining elements has
the same probability of being chosen
...


Example 6
Suppose you have three books A, B and C, but you have room form only two on your
bookshelf
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Example 7
possible? Supposing each of the 26 letters in the alphabet are chosen (with replacement) to be
used, there will be
many possible four letter words that
could be formed
...
10-11)
...
We are no longer interested in ordered samples
just in the samples regardless of the order
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NB:

Example 8
A printed circuit board may be purchased from five suppliers
...
12-13)
...
14):
The number of ways in which elements can be grouped in
and
, is

classes with

in the ith class,

Example 9
In how many different ways can three copies of one novel and one copy each of four other
novels be arranged on a shelf?

Example 10
The number of permutations of the 11 letters of the word MATHEMATICS (which consists

See Examples J

L (Rice, p
...


7

1
...
This concept is
Let A and B be two events with the

...


Multiplication law: Let A and B be events and assume
A and B will occur is:


...
16) and Examples A and B (Rice, p
...

Example 11
Two dice are rolled
...
Suppose that
we rolled the dice one at a time and that the first die came up as a 6 (event B occurred)
...
Then for any event A

For example: Let

B1

B2
A

B3
8

and

See Examples C and D (Rice, p
...

Example 12
A statistics student is randomly chosen at the university
...
5, 0
...
2 respectively
...
4, 0
...
8, for the three years
...

Let A be the event that the student pass and B1 the event that the student is a first year, B2 that
the student is a second year, and B3 that the student is a third year
...
Then

and

(disjoint) for

Example 13 (Continuation of Example 12)
Calculate the probability that a randomly chosen Statistics student is a second year student,
given that he has failed the semester
...
20 - 22)
...
23
Let BC denote the event Breast Cancer and + the event of a positive mammogram
...
Suppose that 20% of the items were produced by M 1, 30% by M2 and
50% by M3
...

i)

Suppose one item is selected at random from the entire batch and it is found to be
defective
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Defective item is denoted by event D
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Determine the probability that it was produced by M2

Example 15
Suppose that a box contains one fair coin and one with a head on each side
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Determine the probability that the coin is the fair coin
...


1
...


or if

Two events A and B are independent if and only if the probability of event B is not
influenced or changed by the occurrence of event A, or vice versa
...
Does the probability that a person is colour-blind change depending on
whether the person is male or not?
Let A be the event that a person is male and B the event that a person is colour blind
...
The probability of event B depends on whether or not event
A has occurred
...

Example 17
Consider tossing a single die two times
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The probability of A:
The probability of B:

Since the probability of B is not changed by the occurrence of event A, we say A and B are
independent
...

Three events A, B and C are independent if and only if:
i)
ii)
iii)
iv)

See Examples A

G (Rice, p
...


Example 18
In a survey of 1000 adults respondents were asked about the expense of college education
...


11

Too high (A)
Child in college 0
...
25
college (E)

Right
(B)
0
...
20

0
...
01

Are events D and A independent?

Therefore A and D are dependent
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12



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