Lecture notes Introduction to Statistics (STAT243)
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Course
Introduction to Statistics (STAT243)
Institution
University Of KwaZulu-Natal (UKZN)
STAT243 is a statistics course that typically covers the following topics:
Descriptive statistics: This includes measures of central tendency (mean, median, mode) and measures of spread (range, standard deviation).
Probability: This includes basic probability theory, such as conditional probabi...
STAT: Data Mining using Machine learning (University of KwaZulu-Natal)
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Chapter 3 – Probability
3.1 Terminology
Probability (Chance)
• A probability is the chance that something of interest will happen.
• A probability is expressed as a proportion i.e. it ranges from 0 to 1.
Chance can be expressed as a percentage i.e. it ranges from 0 to 100.
Examples
1) The probability of rain tomorrow is 0.40
There is a 40% chance of rain tomorrow.
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2) The probability of winning the Lotto is .
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3) The probability of a certain new product being successful is 0.75.
Random experiment
This is an experiment that gives different outcomes when repeated under similar conditions.
1) The experiment can have more than one possible outcome.
2) All possible outcomes can be listed.
3) The outcome that will occur when the experiment is performed depends on
chance.
Examples
1) Tossing a coin (possible outcomes: heads, tails).
2) Rolling a die (possible outcomes: 1, 2, 3, 4, 5, 6).
3) Asking a person to assign a rating to a product (possible outcomes: A, B, C, D, E).
4) Drawing a card from a deck of cards (possible outcomes: 13 hearts, 13 clubs, 13 spades,
13 diamonds).
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Set
A set is a collection of outcomes.
Sample space
The sample space is the set of all possible outcomes of a random experiment. A
sample space is usually denoted by the symbol S and the collection of elements
contained in S enclosed in curly brackets { }.
Sample point
A sample point is an individual outcome (element) in a sample space.
Each outcome listed in the above examples is a sample point.
Event
An event is a subset of a sample space i.e. a collection of sample points taken from a sample
space.
Impossible event
An impossible event is an event that cannot happen (has probability zero).
Certain event
A certain event is an event that is sure to happen (has probability 1).
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Simple events are events that involve only one sample point (outcome) of the sample
space.
Examples
1) Let E denote the event “an odd number is obtained when tossing a single die”.
Then E = {1, 3, 5}.
2) Let H denote the event “at least one head appears when tossing two coins”.
H = {hh, ht, th}.
3) Let B denote the event “obtaining a club and a heart in a single draw from a deck of
cards”. The event B is impossible. The set of outcomes of B is an empty set denoted by
B = { } = φ.
4) Let A denote the event “obtaining a 1, 2, 3, 4, 5 or 6 when tossing a single die”. The
event A is a certain event i.e. one of the outcomes belonging to the set describing the
event must happen. This is denoted by A = S, where S is the sample space.
Venn diagrams
• A Venn diagram is a drawing, in which circular areas represent groups of items
usually sharing common properties.
• The drawing consists of two or more circles, each representing a specific group or
set, contained within a square that represents the sample space. Venn diagrams are
often used as a visual display when referring to sample spaces, events and
operations involving events.
3.2 Complements, Unions and Intersections of
events
Compound events
These are events that involve more than one event. Such events can be obtained by
performing various operations involving two or more events.
Some of the operations that can be performed are described in the sections that follow.
Complementary events
The complementary event Ā (sometimes written À) of an event A is all the outcomes in S
that are not in A.
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