Probability Basics and Mathematical
Text: Chapter 1 and Sections 2.1, 2.2 of Chapter 2
Print version of the lecture in MA340 Probability Theory presented on September 11, 2012
1 Dening a Probability 1
2 Sample Space and Events 2
3 Basic Set Theory 3
3.1 Sets and Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . .
3.2 Operations on Events . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . .
3.3 Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4. . . . . . . .
4 Basic Set Operation Rules 7
5 More on Set Operation Rules: Arbitrary Collections of Sets 7
6 Cartesian Product Sets 8
7 Axioms of Probability 8
8 Probability Space 10 1.2
1 Dening a Probability
Probability theory studies regularities and rules arising in random experiments.
An experiment is called random if its result cannot be predicted in advance with certainty.
Probability Theory gives us the tools to model such events, to understand their linkages, and to
Processes that can be modelled using probability theory are called random processes. Exam-
ples include: the weather, the outcome of a coin toss, lotteries, stock and commodity prices,
Aims of the Course
1 The purpose of MA 340 is to give the student a formal vocabulary of probability models, along
with information about their properties and experience in their application. Besides its direct uses
and extensions, this knowledge complements other areas of pure and applied mathematics, such as
statistics, nancial mathematics, and operations research.
What is a Probability
A chance of how likely it is that some (uncertain) event will occur.
(Frequentist) The probability of an event is the long-run proportion of times that an event E
occurs in a large number n of independent repetitions (trials) of a random experiment:
A common example is the "repeated coin toss experiment" where the ratio of the number of H
(heads) to T (tails) converges to the true probability of tossing H or T, e.g. for a fair coin this
prob. is 1=2.
(Classical) An events probability is the ratio of the number of favourable outcomes and all
possible outcomes in a random experiment:
P(E) = n(E)
where W is a nite sample space of equally likely outcomes.
A probability is a measure of likelihood on a scale of 0 to 1. The probability of an impos-
sible event is zero, the probability of an inevitable (certain) event is one. A rare event has a
probability close to zero. A very common event has a probability close to one.
A population is a collection of all individuals or items (objects) under consideration. If a
member is selected at random from a nite population, then the probability that the member
has a specic attribute equals the percentage (or proportion) of the population that has that
2 Sample Space and Events
A (random) experiment is a phenomenon where the outcome cannot be predicted with cer-
tainty in advance.
A sample space is the set of all possible outcomes of the experiment. It is denoted by the
Greek letter W.
Note that every outcome of the experiment is described by one and only one, sample point. A
possible outcome is denoted by w.
A sample space is discrete if it consists of a nite or countable innite set of outcomes, and is
continuous if it contains an interval (either nite or innite) of real numbers.
Any subset of a sample space, i.e. any part of the set, including the whole set and an empty set,
is called an event.
We say that an event E occurs if the outcome of a random experiment belongs to the event:
w 2 E.
A probability is a number between 0 and 1 assigned to an event
2Historical Examples of Random Experiments
The naturalist Buffon tossed a coin 4040 times, resulting in 2048 heads and 1992 tails.
He also estimated the number p by throwing needles on a ruled surface and recording how
many times the needles crossed a line
English biologist W. F. R. Weldon recorded 26,306 throws of 12 dice, and the Swiss scientist
Rudolf Wolf recorded 100,000 throws of a single die without a computer
The statistician Karl Pearson analyzed a large number of outcomes at certain roulette tables
and suggested that the wheels were biased.
In 1955, RAND Corporation printed a table of 1,000,000 random numbers generated from
Examples of sample spaces
Experiment Outcome Sample Space W
tossing a coin Head or Tail fH;Tg
rolling a die the number of f1;2;3;4;5;6g