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JSE Company Data Probability Analysis

This document presents a comprehensive analysis of company data from the Johannesburg Stock Exchange (JSE), detailing sector distributions, probability calculations for various company sizes and sectors, and summarizing key findings.

Prompt

One hundred and seventy (170) companies from the JSE were randomly selected and classified 
by sector and size. The table below shows the frequencies for the two categorical random 
variables, ‘sector’ and ‘size’. 
Sector 
Company Size 
Small 
Row Total 
Medium 
Large 
Mining 
3 
8 
30 
Financial 
41 
9 
21 
42 
Service 
72 
10 
6 
8 
Retail 
24 
14 
13 
6 
Column Total 
33 
36 
48 
86 
170 
a. What is the probability that a randomly selected JSE company will be small and operate in the 
service sector?                                                                           
(3 marks) 
b. What is the probability that a randomly selected JSE company will be both small and medium
sized?                   
(3 marks) 
c. What is the probability that a randomly selected JSE company will be either a small company, 
a service sector company, or both?               
(3 marks) 
d. What is the probability that a randomly selected company is a retail company, given that it is 
known (in advance) to be a medium-sized company?             
(3 marks) 
e. What is the probability of selecting a small retail company from the JSE-listed sample 
companies?

Answer

Analysis of JSE Company Data

Given Data:

Total companies = 170

Sector	Small	Medium	Large	Row Total
Mining	3	8	30	41
Financial	41	9	21	71
Service	72	10	6	88
Retail	24	14	13	51
Column Total	33	36	48	170

Probability Calculations

a. Probability of Small and Service Sector

Formula:
P(Small ∩ Service) = (Number of Small Service Companies) / (Total Companies)

Calculation:
P(Small ∩ Service) = 72 / 170 = 0.4242

b. Probability of Small and Medium Sized

Formula:
P(Small ∩ Medium) = (Number of Small and Medium Companies) / (Total Companies)

Calculation:
P(Small ∩ Medium) = 0 / 170 = 0.0

c. Probability of Small or Service Sector

Formula:
P(Small ∪ Service) = P(Small) + P(Service) - P(Small ∩ Service)

Calculations:

P(Small) = (Total Small Companies) / (Total Companies) = 33 / 170 = 0.1941
P(Service) = (Total Service Companies) / (Total Companies) = 88 / 170 = 0.5176
P(Small ∩ Service) = 72 / 170 = 0.4242

Putting it together:
P(Small ∪ Service) = 0.1941 + 0.5176 - 0.4242 = 0.2875

d. Probability of Retail Given Medium Size

Formula:
P(Retail | Medium) = P(Retail ∩ Medium) / P(Medium)

Calculations:

P(Retail ∩ Medium) = (Number of Medium Retail Companies) / (Total Companies) = 14 / 170
P(Medium) = (Total Medium Companies) / (Total Companies) = 36 / 170

Thus,
P(Retail | Medium) = (14 / 170) / (36 / 170) = 14 / 36 = 0.3889

e. Probability of Selecting a Small Retail Company

Formula:
P(Small ∩ Retail) = (Number of Small Retail Companies) / (Total Companies)

Calculation:
P(Small Retail) = 24 / 170 = 0.1412

Summary of Probabilities

Scenario	Probability
Small and Service Sector	0.4242
Small and Medium Sized	0.0
Small or Service Sector	0.2875
Retail Given Medium	0.3889
Small Retail Company	0.1412

Conclusion

The above calculations represent the probabilities of various scenarios based on the provided company data from JSE. Each scenario has been derived using fundamental principles of probability, ensuring clarity and correctness in the analysis.