Question 4.4: JSE Companies – Sector and Company Size Study Refer to Table......

(a) The ‘Column total’ row in Table 4.3 describes the outcomes for the random variable ‘company size’ only and is therefore used to find marginal probabilities for the variable ‘company size’.

Let A = event (large company).
From Table 4.3, there are 86 large companies in the sample of 170 companies.

Thus P(A) =\frac{86}{170} = 0.506 (50.6%).

There is a 50.6% chance that a randomly selected JSE company will be large.

(b) The ‘Row total’ column in Table 4.3 describes the outcomes for the random variable ‘sector’ only and is therefore be used to find marginal probabilities for the variable ‘sector’.

Let B = event (mining company).

From Table 4.3, there are 41 mining companies in the sample of 170 companies.

Thus P(B) = \frac{41}{170} = 0.241 (24.1%).

There is a 24.1% chance that a mining company will be randomly selected.

(c) Let A = event (medium-sized company).
Let B = event (financial company).

From Table 4.3, there are 21 medium-sized financial companies in the study. Thus P(A ∩ B) = \frac{21}{170} =0.124 (12.4%).

There is a 12.4% chance of randomly selecting a medium-sized financial company.

(d) Let A = event (retail company).
Let B = event (medium-sized company).
Then P(A|B) = P(retail company|medium-sized company).

The sample space is reduced to the subset of 48 medium-sized companies only (as shown in Table 4.4). Within this sub-sample of 48 medium-sized companies, there are 13 retail companies.

Thus P(A|B) = P(retail|medium) = \frac{13}{48} = 0.271 (27.1%).

This same result can be found by applying Formula 4.2 as follows:

• P(B) = P(medium-sized) = \frac{48}{170} = 0.2824 (marginal probability).
• P(A ∩ B) = P(retail ∩ medium-sized) = \frac{13}{170} = 0.0765 (joint probability).
• Then P(A|B) = \frac{\frac{13}{170} }{\frac{48}{170} }=\frac{13}{48} = 0.271 (conditional probability).

There is a 27.1% chance that a randomly selected JSE company will be a retail company, given that it is known (in advance) to be a medium-sized company.