Condition tional Probabil al Probability P(A ity P(A ||B B))
4.6 Probability Probability Rules Rules
4.6 Probability Probability Rules Rules
Probability rules have been developed to calculate probabilities of compound or multiple events occurring simultaneously. There are two basic probability rules:
Theaddition ruleaddition rule
– for non-mutually exclusive events, and – for mutually exclusive events.
The addition rule relates to the union of events. It is used to find the proba bility of either event Aor event B,or both events occurring simultaneously in a single trial of a random experiment.
Themultiplication rulemultiplication rule
– for statistically dependent events, and – for statistically independent events.
The multiplication rule relates to the intersection of events. It is used to find the probability of event Aand event B occurring together in a single trial of a random experiment.
Refer to Table 4.5, which illustrates the application of these probability rules.
Addition Rule for Non-Mutually Exclusive Events
If two events arenotmutually exclusive, theycanoccur together in a single trial of a random experiment. Then the probability ofeither event A or event B orboth occurring in a single trial of a random experiment is defined as:
Chapter 4 – Basic Probability Concepts Chapter 4 – Basic Probability Concepts
P(A∪ B) = P(A) + P(B) − P(A∩B) 4.34.3
In Venn diagram terms, the union of two non-mutually exclusive events is thecombined outcomes of the two overlapping events A and B – as described in Concept 2 and shown graphically in figures 4.3 and 4.4.
Example 4.5
Example 4.5 JSE Companies JSE Companies – Sector and – Sector and Size StudySize Study Table 4.5
Table 4.5 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
(a) What is the probability that a randomly selected JSE-listed company iseither a large companyor a financial company,or both?
Solution Solution
(a) Let A = event (large company).
Let B = event (financial company).
Events A and Bare not mutually exclusive as they can occur simultaneously (i.e. a company can beboth large and financial).
From Table 4.5, the following marginal and joint probabilities can be derived:
P(A) = P(large) =
___
86170= 0.5059 P(B) = P(financial) =
___
72170= 0.4235 P(A∩B) = P(large and financial) =
___
42170= 0.2471 Then P(A ∪ B) = P (either large or financial orboth)
= P(large) + P( financial) − P(large and financial)
= 0.5059 + 0.4235 – 0.2471
= 0.682 (68.2%)
There is a 68.2% chance that a randomly selected JSE-listed company will beeither a large companyor a financial company,or both (i.e. a large financial company).
Addition Rule for Mutually Exclusive Events
If two events are mutually exclusive, theycannot occur together in a single trial of a random experiment. Then the probability ofeither event Aor event B (butnot both) occurring in a single trial of a random experiment is defined as:
Applied Business Statistics
P(A∪ B) = P(A) + P(B) 4.44.4
i.e. it is the sum of only the two marginal probabilities of events A and B.
In Venn diagram terms, the union of two non-mutually exclusive events is thesum of the outcomes of each of the two (non-overlapping) events A and Bseparately – as described in Concept 3 and shown graphically in Figure 4.5.
For mutually exclusive events, there isnointersectional event. Thus P(A∩ B) = 0.
Example 4.5 contd.
Example 4.5 contd.
(b) What is the probability that a randomly selected JSE-listed company is either a mining company or a service company?
Solution Solution
Let A = event (mining company).
Let B = event (service company).
Events A and B are mutually exclusive as they cannot occur simultaneously (a company cannot be both a mining company and a service company), so P(A∩ B) = 0.
From Table 4.5, the following marginal probabilities can be derived:
P(A) = P(mining) =
___
41170= 0.241 (24.1%) P(B) = P(service) =
___
24170= 0.141 (14.1%) P(A∩ B) = 0
Then P(A∪ B) = P(either mining or service)
= P(mining) + P(service)
= 0.241 + 0.141
= 0.382 (38.2%)
There is a 38.2% chance that a randomly selected JSE-listed company will be either a mining company or a service company, but not both.
Multiplication Rule for Statistically Dependent Events
The multiplication rule is used to find the joint probability of events A and B occurring together in a single trial of random experiment (i.e. theintersection of the two events). This rule assumes that the two events A and B are associated (i.e. they are dependent events).
The multiplication rule for dependent events is found by rearranging the conditional probability formula (Formula 4.2), resulting in:
P(A∩ B) = P(A|B) × P(B) 4.54.5
Where P(A ∩ B) = joint probability of A and B P(A|B) = conditional probability of A given B P(B) = marginal probability of B only Example 4.5 contd.
Example 4.5 contd.
(c) What is the probability of selecting a small retail company from the JSE-listed sample of companies?
Chapter 4 – Basic Probability Concepts Chapter 4 – Basic Probability Concepts
Solution Solution
Let A = event (small company).
Let B = event (retail company).
Intuitively, from Table 4.5, P(A∩ B) = P(small and retail) =
___
14170= 0.082 (8.2%).
Alternatively, this probability can be calculated from the multiplication rule Formula 4.5.
P(B) = P(retail) =
___
33170= 0.1942 P(A|B) = P(small|retail) = ___ 14 33= 0.4242 Then P(A∩ B) = P(A|B) × P(B)
= P(small|retail) × P(retail)
= 0.4242 × 0.1942
= 0.082 (8.2%)
There is only an 8.2% chance that a randomly selected JSE-listed company will be a small retail company.
Multiplication Rule for Statistically Independent Events
If two events A and B are statistically independent (i.e. there is no association between the two events) then the multiplication rule reduces to the product of the two marginal probabilities only.
P(A∩ B) = P(A) × P(B) 4.64.6
Where: P(A∩ B) = joint probability of A and B P(A) = marginal probability of A only P(B) = marginal probability of B only
The following test can be applied to establish if two events are statistically independent. Two events are statistically independent if the following relationship is true:
P(A|B) = P(A) 4.74.7
This means that if themarginal probability of event Aequals theconditional probability of event Agiven that event B has occurred, then the two events A and B are statistically independent.
This implies that the prior occurrence of event B inno way influences the outcome of event A.
Example 4.5 contd.
Example 4.5 contd.
(d) Is company size statistically independent of sector in the JSE-listed sample of companies?
Solution Solution
To test for statistical independence, select one outcome from each event A and B and apply the decision rule in Formula 4.7 above.
Let A = event (medium-sized company).
Let B = event (mining company).
Then P(A) = P(medium-sized company) =
___
48170= 0.2824 (28.24%)
P(A|B) = P(medium-sized company|mining) = ___ 418 = 0.1951 (19.51%)
Applied Business Statistics
Since the two probabilities are not equal (i.e. P(A|B) ≠ P(A)), the empirical evidence indicates that company size and sector are statistically dependent (i.e. they are related).