QUAT6221 LU5
QUAT6221 LU5 – Basic Probability
Concepts
Chapter 4
4.1 Intro
A probability is the chance or likelihood that a particular event will occur.
4.2 Types of probability
2 types
1. Subjective
Probability is based on an educated guess, expert opinion or intuition
2. Objective
Probability can be verified statistically
Probability = ratio of 2 numbers (math definition)
r
P ( A )=
n
A = event of a specific type
r = number of outcomes of event A
n = total number of all possible outcomes / sample space
P(A) = probability of event A happening
Example 4.1
Assume that 355 Ford car owners were randomly selected and asked the following question:
'When you buy your next car, will you buy another Ford product?
Assume that 76 respondents answered 'Yes'
76
P ( A )=
355
P(A) = 0.214….
There is a 21.4% chance that a current Ford owner will by another Ford for their next car.
Deriving Objective Probabilities
There are 3 ways objective probabilities are derived:
1. A priori – outcomes are known in advance
2. Empirically – values of r and n are unknown but can be observed through data
collection
3. Mathematically – using formulas to calculate probabilities
1
, QUAT6221 LU5
4.3 Properties of a probability
5 basic properties that apply to every probability:
1. The values lies between 0 and 1 inclusive
2. If an event cant occur P(A) = 0
3. If an event is certain to occur P(A) = 1
4. The sum of probabilities of all possible events equals 1
P(A1) + P(A2) + P(A3) + … P(Ak) = 1 for k possible events
5. Complementary probability
If P(A) is the probability of event A happening, then the probability of event A
not happening is P( A ) = 1 – P(A)
4.4 Basic Probability Concepts
The following concepts are relevant when calculating probabilities associated w 2 or more
events occurring:
- Intersection of events ∩
o Set of all outcomes that belongs to both events simultaneously “and”
o Venn diagrams!
- Union of events ∪
o Set of all outcomes that belong to either event “or”
- Mutually exclusive events
o Events that cannot occur together / not at the same point in time
- Collectively exhaustive events
o Union of all possible events is equal to the sample space
- Statistically independent events
o If the occurrence of event A has no effect on the outcome of event B and vice
versa
o Can occur together but have no effect on each other
Example 4.3
170 companies from the JSE were randomly selected and classified by sector and size.
Table 4.2 shows the cross-tabulation table of joint frequencies for the two categorical
random variables 'sector' and 'company size'.
Sector Company size Row Total
Small Medium Large
Mining 3 8 30 41
Financial 9 21 42 72
Service 10 6 8 24
Retail 14 13 6 33
Column Total 36 48 86 170
1. What is the probability of a randomly selected company will be small and operate in
the service sector?
Let A = event (small company) Let B = event (service company)
The A∩B is the set of all small and service companies.
2
QUAT6221 LU5 – Basic Probability
Concepts
Chapter 4
4.1 Intro
A probability is the chance or likelihood that a particular event will occur.
4.2 Types of probability
2 types
1. Subjective
Probability is based on an educated guess, expert opinion or intuition
2. Objective
Probability can be verified statistically
Probability = ratio of 2 numbers (math definition)
r
P ( A )=
n
A = event of a specific type
r = number of outcomes of event A
n = total number of all possible outcomes / sample space
P(A) = probability of event A happening
Example 4.1
Assume that 355 Ford car owners were randomly selected and asked the following question:
'When you buy your next car, will you buy another Ford product?
Assume that 76 respondents answered 'Yes'
76
P ( A )=
355
P(A) = 0.214….
There is a 21.4% chance that a current Ford owner will by another Ford for their next car.
Deriving Objective Probabilities
There are 3 ways objective probabilities are derived:
1. A priori – outcomes are known in advance
2. Empirically – values of r and n are unknown but can be observed through data
collection
3. Mathematically – using formulas to calculate probabilities
1
, QUAT6221 LU5
4.3 Properties of a probability
5 basic properties that apply to every probability:
1. The values lies between 0 and 1 inclusive
2. If an event cant occur P(A) = 0
3. If an event is certain to occur P(A) = 1
4. The sum of probabilities of all possible events equals 1
P(A1) + P(A2) + P(A3) + … P(Ak) = 1 for k possible events
5. Complementary probability
If P(A) is the probability of event A happening, then the probability of event A
not happening is P( A ) = 1 – P(A)
4.4 Basic Probability Concepts
The following concepts are relevant when calculating probabilities associated w 2 or more
events occurring:
- Intersection of events ∩
o Set of all outcomes that belongs to both events simultaneously “and”
o Venn diagrams!
- Union of events ∪
o Set of all outcomes that belong to either event “or”
- Mutually exclusive events
o Events that cannot occur together / not at the same point in time
- Collectively exhaustive events
o Union of all possible events is equal to the sample space
- Statistically independent events
o If the occurrence of event A has no effect on the outcome of event B and vice
versa
o Can occur together but have no effect on each other
Example 4.3
170 companies from the JSE were randomly selected and classified by sector and size.
Table 4.2 shows the cross-tabulation table of joint frequencies for the two categorical
random variables 'sector' and 'company size'.
Sector Company size Row Total
Small Medium Large
Mining 3 8 30 41
Financial 9 21 42 72
Service 10 6 8 24
Retail 14 13 6 33
Column Total 36 48 86 170
1. What is the probability of a randomly selected company will be small and operate in
the service sector?
Let A = event (small company) Let B = event (service company)
The A∩B is the set of all small and service companies.
2
