Deriving Objective ProbabilitiesDeriving Objective Probabilities
4.4 Basic Basic Probabil Probability ity Concepts Concepts
4.4 Basic Basic Probabil Probability ity Concepts Concepts
The following concepts are relevant when calculating probabilities associated with two or more events occurring:
theintersection of events theunion of events
mutually exclusive events collectively exhaustive events statistically independent events.
These basic probability concepts will be illustrated using the following example:
Example 4.3
Example 4.3 JSE Companies JSE Companies – Sector and – Sector and Size StudySize Study
One hundred and seventy (170) companies from the JSE were randomly selected and classified by sector and size. Table 4.2 shows the cross-tabulation tablecross-tabulation table of joint frequencies for the two categorical random variables ‘sector’ and ‘company size’.
(SeeExcel file C4.2 – jse companies.)
Applied Business Statistics
Table 4.2
Table 4.2 Cross-tabulation table – JSE companies by sector and size SSeeccttoorr CCoommppaanny y ssiizzee RRoow w ttoottaall
SSmmaallll MMeeddiiuumm LLaarrggee
Mining 3 8 30 41
Financial 9 21 42 72
Service 10 6 8 24
Retail 14 13 6 33
Column total
Column total 36 48 86 170
These frequency counts are used to deriveempirical probabilitiesempirical probabilities, since the data was gathered from a survey and organised into a summary table.
Concept 1: Intersection
Concept 1: Intersection of Tof Two Events wo Events (A(A∩ B) B)
Theintersectionintersection of two events A and B is the set of all outcomes that belong to both A and Bsimultaneously. It is written as A∩ B (i.e. A and B), and the keyword is ‘andand’.
Figure 4.1 shows the intersection of events graphically, using a Venn diagram. The intersection of two simple events in a Venn diagram is called a joint ev joint eventent.
Sample space =n
A A∩ B B
Figure 4.1
Figure 4.1 Venn diagram of the intersection of two events (A∩ B)
(a) What is the probability that a randomly selected JSE company will be small and operate in the service sector?
Solution Solution
(a) Let A = event (small company).
Let B = event (service sector company).
Then (A ∩ B) is the set of all smalland service sector companies.
From Table 4.2, there are 10 companies that are both small and operate in the service sector, out of 170 JSE companies surveyed. This is shown graphically in the Venn diagram in Figure 4.2 below.
Thus P(A ∩ B) = P(small∩ service) =
___
10170= 0.0588.
There is only a 5.9% chance of selecting a small service sector JSE company.
Chapter 4 – Basic Probability Concepts Chapter 4 – Basic Probability Concepts
n= 170 JSE Companies Small
only
Service only Small
and service
(10)
Figure 4.2
Figure 4.2 Venn diagram of small and service JSE companies (intersection) Concept 2: Union of T
Concept 2: Union of Two Events wo Events (A(A∪ B) B)
Theunionunion of two events A and B is the set of all outcomes that belong toeither event A or B or both. It is written as A∪ B (i.e. either Aor B or both) and the key word is ‘oror’.
Figure 4.3 shows the union of events graphically using a Venn diagram.
Sample space =n
A B
A∪ B
Figure 4.3
Figure 4.3 Venn diagram of the union of events (A∪ B)
(b) What is the probability that a randomly selected JSE company will beeither a small companyor a service sector company,or both?
Solution Solution
(b) Let A = event (small company).
Let B = event (service company).
Then (A∪ B) is the set of all smallor serviceor both (small and service) companies.
As seen in Table 4.2, there are 36 small companies (includes 10 service companies), 24 service companies (includes 10 small companies) and 10 small and service companies. Therefore, there are 50 separate companies (36 + 24 − 10) that are either smallor service,or both. Note that the intersection (joint) event is subtracted once to avoid double counting. This is shown in the Venn diagram in Figure 4.4 below.
Thus P(A∪ B) = P(small∪ service) = __________ 36 + 24 – 10 170 =
___
50170= 0.294.
There is a 29.4% chance of selectingeither a smallor a service JSE company,or both.
Applied Business Statistics
n =170 Small
(36)
Service (24) Small
and service
(10)
Figure 4.4
Figure 4.4 Venn diagram of small or service JSE companies (union) Concept 3 Mutually Exclusive Events
Concept 3 Mutually Exclusive Events
Concept 3: Mutually Exclusive and Non-mutually Exclusive EventsEvents aremutually exclusivemutually exclusive if theycannot occur together on a single trial of a random experiment (i.e. not at the same point in time).
Figure 4.5 graphically shows events that are mutually exclusive (i.e. there is no intersection) using a Venn diagram.
Sample space =n A
B
Figure 4.5
Figure 4.5 Venn diagram of mutually exclusive events, A∩B = 0
(c) What is the probability of a randomly selected JSE company being both a small and a medium-sized company?
Solution Solution
(c) Let A = event (small company).
Let B = event (medium company).
Events A and B aremutually exclusive, since a randomly selected companycannot be both small and medium at the same time.
Thus P(A ∩ B) = P(small∩ medium) = 0 (i.e. the joint event is null).
There is no chance of selecting a small- and medium-sized JSE company simultaneously. It is therefore an impossible event.
Events arenon-mutually exclusive if they canoccur together on a single trial of a random experiment (i.e. at the same point in time). Figure 4.1 graphically shows events that are non-mutually exclusive (i.e. there is an intersection). Example (a) above illustrates probability calculations for events that are not mutually exclusive.
Chapter 4 – Basic Probability Concepts Chapter 4 – Basic Probability Concepts
Concept 4: Collectively Exhaustive EventsConcept 4: Collectively Exhaustive Events
Events arecollectively exhaustivecollectively exhaustive when the union of all possible events is equal to the sample space.
This means, that in a single trial of a random experiment, at least one of these events is certain to occur.
(d) What is the probability of selecting a small, medium or large JSE company from the sample of 170 companies surveyed?
Solution Solution
(d) Let A = event (small company).
Let B = event (medium company).
Let C = event (large company).
Since (A ∪ B∪ C) = (the sample space of all JSE companies) then P(A ∪ B∪ C) = P(small) + P(medium) + P(large)
=
___
36170+
___
48170+
___
86170
= 0.212 + 0.282 + 0.506 = 1
Since the events comprise the collectively exhaustive set for all company sizes, the event of selecting either a small or medium or large JSE company iscertain to occur .
Concept 5: Statistically Independent Events Concept 5: Statistically Independent Events
Two events, A and B, arestatistically independentstatistically independent if the occurrence of event A has no effect on the outcome of event B, andvice versa.
For example, if the proportion of male clients of Nedbank who use internet bankin g is the same as the proportion of Nedbank’s female clients who use internet banking, then
‘gender’ and ‘preference for internet banking’ at Nedbank are statistically independent events.
A test for statistical independence will be given in section 4.6.
A word of caution
The terms ‘statistically independent’ events and ‘mutually exclusive’ events are often confused. They are two very different concepts. The distinction between them is as follows:
When two events are mutually exclusive, they cannot occur together.
When two events are statistically independent, they can occur together, but they do not have an influence on each other.