Unit 3 - BASICS OF
Unit 3 - BASICS OF
PROBABILITY
PROBABILITY
Definitions
What is Probability?
Definitions I
An OutcomeOutcome
The result/observation from an experimentThe result/observation from an experiment
The Sample SpaceSample Space (S)
The collection of all outcomes from an experiment
Illustrated using venn diagram or a tree diagram
An EventEvent
Is a collection of one or more of the outcomes of an
Definitions II
A setset is a collection of elements or objects of interest
Empty set (denoted by )
a set containing no elements
Universal Set (denoted by X or S)
a set containing all possible elements
Definitions II
Complement (Not)
denoted by Ac, or ).
The complement of A is a set containing all elements NOT in A.
Definitions II
Intersection Intersection
(And)
a set containing all elements in both A and B
Joint occurrence
Happening at the same time
Simultaneously
Both
A AB B
Definitions II
UnionUnion (Or)
a set containing all
elements in either A, or B or both A and B
Either Or
Definitions II
Mutually ExclusiveMutually Exclusive Events
sets having no elements in common, having no intersection.
Intersection is the empty set. Cannot happen at the same time
Example – All Types of
Events
Let the sample space be the collection of all possible outcomes of rolling one
die:
S = [1, 2, 3, 4, 5, 6]
Let A be the event “Number rolled is even”
Let B be the event “Number rolled is at least 4” Then:
A = [2, 4, 6] and B = [4, 5, 6]
What elements are in:
1. The complement of A 2. The complement of B 3. A intersect B
4. A union B
DeMorgan’s Law
Augustus DeMorganDeMorgan originally observed that:
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and
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A
B
A
B
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DeMorgan’s Law
Let A = Statistics is boring
Let B = Mathematics is interesting
Translation:
LHS = It is not true that Statistics is boring OR Mathematics is
interesting
RHS = Statistics is not boring AND Mathematics is not interesting
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DeMorgan’s Law
Translation:
LHS = It is not true that Statistics is boring AND
Mathematics is interesting
RHS = Statistics is not boring OR Mathematics is not
interesting
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Example – Venn Diagrams
Use Venn Diagrams to illustrate the following:
1. A∩B’
2. A ∪ B’
3. (A ∪ B)’
What is Probability?
Probability is a numerical measure of the likelihood
that a specific event will occur.
Properties of Probability
Expressing Probability
The formula to calculate probability is:
Probability can be expressed in three ways:
Fraction Proportion Percentage
outcomes possible
of #
outcomes favourable
Calculating Probability
Example 1:
Find the probability of
obtaining an even
number in one roll of a die
Example 2:
In a group of 500
women, 120 have played golf at least once. Suppose one of those 500 women is
Probability - Complement
P(A) means "Probability of Event A”
P(A') means "Probability of the complement of Event A"
Example - Complement
In a group of 200 tax payers, 40 have been
audited by the IRS at least once. If one tax payer is randomly selected from the group, what is the probability:
1. Selected tax payer has been audited
Probability - Independent
Events
Events are considered independent if the
occurrence of one does not affect the occurrence of the other
Example – Independent
Events I
Example – Independent
Events II
The probability that a salesman will make a sale if he has an appointment is 0.3. The salesman
has two appointments on a certain day.
Assuming that the sales are independent, determine the probability that:
a. he will make two sales;
b. he will make exactly one sale;
Probability – Union &
Intersection
For independent events,
Example – Union &
Intersection
1. In a group of 250 persons, 140 are female, 60 are
vegetarians and 40 are females and vegetarian. What is the probability that a randomly selected person
from this group is a male or vegetarian?
2. Jason and Lisa are planning an outdoor wedding
Probability – Mutually
Exclusive Events
If A and B are mutually exclusive, then P(A B) = P(A) + P(B)∪
Example – Mutually
Exclusive Event
1. What is the probability of a die showing a 2 or a 5?
2. The probabilities of three teams A, B and C winning a
badminton competition are 1/3, 1/5 and 1/9 respectively. Calculate the probability that:
a) either A or B will win
b) either A or B or C will win
c) none of these teams will win
Marginal, Joint and
Conditional Probability
Marginal (Simple) Probability
The probability of a single event without consideration of any other event
Joint Probability
Marginal, Joint and
Conditional Probability
Conditional Probability
The is the probability of event A occurring, given that event B has already occurred
Denoted by P(A|B)
Read: the probability of A given B
Conditional Probability Formula: ( )
Conditional Probability &
Independence
If and only if (iff) the events are independent, then the conditional probability is equal to
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A
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Contingency Table
Table shows
distribution of 200 employees based on two characteristics
1. Gender (male or female) 2. Promotion
Each cell gives the frequency for two characteristics
Male Female Total
Promoted 80 40 120
Not
Promoted 30 50 180
Marginal Probability
How many
persons were promoted?
What is the
probability of being promoted?
Male Female Total
Promoted 80 40 120
Not
Promoted 30 50 180
Joint Probability
How many males
were promoted?
What is the
probability of being male and promoted?
Male Female Total
Promoted 80 40 120
Not
Promoted 30 50 180
Conditional Probability
Does promotion
appear to be
related to gender? Explain.
Male Female Total
Promoted 80 40 120
Not
Promoted 30 50 180
Example –
Marginal/Joint/Conditional
If a person is selected at random, what is the probability that he gets
a dealer who provides good service?
If a person randomly selects a dealer, what is the probability that he
gets a dealer who was in business for less than 10 years and also provides good service?
If a person randomly selects a dealer who was in business for more
than 10 years, what is the probability that he gets one that provides good service?
Years in Business
Quality of Service After
Warranty Total
Good Bad
≥ 10 16 4 20
