Question 4.3: JSE Companies – Sector and Size Study One hundred and sevent......

(a) Let A = event (small company).
Let B = event (service sector company).
Then (A ∩ B) is the set of all small and 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.

Thus P(A ∩ B) = P(small ∩ service) = \frac{10}{170} = 0.0588.

There is only a 5.9% chance of selecting a small service sector JSE company.

Concept 2: Union of Two Events (A ∪ B)

The union of two events A and B is the set of all outcomes that belong to either event A or B or both. It is written as A ∪ B (i.e. either A or B or both) and the key word is ‘or’.

Figure 4.3 shows the union of events graphically using a Venn diagram.

(b) Let A = event (small company).
Let B = event (service company).
Then (A ∪ B) is the set of all small or service or 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 small or 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) = \frac{36+24-10}{170}=\frac{50}{170}=0.294.

There is a 29.4% chance of selecting either a small or a service JSE company, or both.

Concept 3 Mutually Exclusive Events

Events are mutually exclusive if they cannot 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.

(c) Let A = event (small company).
Let B = event (medium company).
Events A and B are mutually exclusive, since a randomly selected company cannot 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 are non-mutually exclusive if they can occur 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.

Concept 4: Collectively Exhaustive Events

Events are collectively 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) 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)

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 is certain to occur.

Concept 5: Statistically Independent Events

Two events, A and B, are statistically independent if the occurrence of event A has no effect on the outcome of event B, and vice versa.

For example, if the proportion of male clients of Nedbank who use internet banking 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.