Download Definition

Chapter 1: Random Events and
Probability
Department of Statistics
Huang Xudong,Ph.D
§1.1 Random event
1.1.1 Random Experiments
The basic notion in probability is that of a random
experiment:
an experiment whose outcome cannot be
determined in advance, but is nevertheless still
subject to analysis.
Examples of random experiments are:





1. tossing a die,
2. measuring the amount of rainfall in Brisbane in
January,
3. counting the number of calls arriving at a
telephone exchange during a fixed time period,
4. selecting a random sample of fifty people and
observing the number of left-handers,
5. choosing at random ten people and measuring
their height.
1.1.2 Sample Space

Definition The sample space Ω of a random
experiment is the set of all possible outcomes of
the experiment.
Examples of random experiments with
their sample spaces are:
1. Cast two dice consecutively,
2. The lifetime of a machine (in days),
3. The number of arriving calls at an exchange during
a specified time interval,
4. The heights of 10 selected people.
Discrete and continuous sample spaces




Definition: A sample space is finite if it has a
finite number of elements.
Definition: A sample space is discrete if there are
“gaps” between the different elements, or if the
elements can be “listed”, even if an infinite list
(eg. 1, 2, 3, . . .).
In mathematical language, a sample space is
discrete if it is countable.
Definition: A sample space is continuous if there
are no gaps between the elements,so the elements
cannot be listed (eg. the interval [0, 1]).
1.1.3 Events
So far, we have introduced the sample space,
Ω,which lists all possible outcomes of a random
experiment, and might seem unexciting.
However, Ω is a set. It lays the ground for a whole
mathematical formulation of randomness, in terms
of set theory.
The next concept that you would need to
formulate is that of something that happens at
random, or an event.
How would you express the idea of an event in
terms of set theory?
Definition of events



Definition: An event is a subset of the sample space.
That is, any collection of outcomes forms an event.
Events will be denoted by capital letters A,B,C,....
Note:We say that event A occurs if the outcome of
the experiment is one of the elements in A.
Note: Ω is a subset of itself, so Ω is an event. The
empty set, ∅ = {}, is also a subset of Ω. This is called
the null event, or the event with no outcomes.
Examples of events are:
1. The event that the sum of two dice is 10 or more,
2. The event that a machine lives less than 1000
days,
3. The event that out of fifty selected people, five are
left-handed,
Combining Events


Formulating random events in terms of sets gives us
the power of set theory to describe all possible ways
of combining or manipulating events. For example,
we need to describe things like coincidences (events
happening together), alternatives, opposites, and so
on.
We do this in the language of set theory.
Example: Suppose our random experiment is to
pick a person in the class and seewhat form(s) of
transport they used to get to campus today.

This sort of diagram representing events in a
sample space is called a Venn diagram.
1. Alternatives: the union ‘or’ operator


Definition: Let A and B be events on the same
sample space Ω: so A⊂Ω and B⊂Ω.
The union of events A and B is written A∪B, and
is given by
2. Concurrences and coincidences: the
intersection ‘and’ operator

Definition: The intersection of events A and B is
written A ∩B and is given by
3. Opposites: the complement or ‘not’
operator

Definition: The complement of event A is written
and is given by
Examples:



Experiment: Pick a person in this class at random.
Sample space: Ω = {all people in class}.
Let event A =“person is male” and
event B = “person travelled by bike today”.
Suppose I pick a male who did not travel by bike.
Say whether the following events have occurred:
Properties of union, intersection, and
complement
Distributive laws
1.1.4 Partitioning sets and events
Examples:
Partitioning an event A
§1.2 Frequency and probability
1.2.1 Frequency
Consider performing our experiment a large number n
times and counting the number of those times when A
occurs. The relative frequency of A is then defined to
be
f n ( A)  n A n
When n A is the number of times that A occurs.
Properties of frequency:
1.2.2 Probability: a way of measuring sets


Remember that you are given the job of building
the science of randomness. This means somehow
‘measuring chance’.
It was clever to formulate our notions of events
and sample spaces in terms of sets: it gives us
something to measure. ‘Probability’, the name
that we give to our chance-measure, is a way of
measuring sets.
Most of this course is about probability
distributions.


A probability distribution is a rule according to
which probability is apportioned,or distributed,
among the different sets in the sample space.
At its simplest, a probability distribution just lists
every element in the sample space and allots it a
probability between 0 and 1, such that the total
sum of probabilities is 1.
Discrete probability distributions
Continuous probability distributions


On a continuous sample space Ω, e.g. Ω = [0, 1],
we can not list all the elements and give them an
individual probability. We will need more
sophisticated methods detailed later in the course.
However, the same principle applies. A continuous
probability distribution is a rule under which we
can calculate a probability between 0 and 1 for any
set, or event, A ⊆ Ω.
1.2.3 Probability Axioms

For any sample space, discrete or continuous, all
of probability theory is based on the following
three definitions, or axioms.
If our rule for ‘measuring sets’ satisfies the three
axioms, it is a valid probability distribution.

Note: The axioms can never be ‘proved’: they are
definitions.
Note: Remember that an EVENT is a SET: an
event is a subset of the sample space.
1.2.3 Probabilities of combined events

In Section 1.3 we discussed unions, intersections,
and complements of events. We now look at the
probabilities of these combinations. Everything
below applies to events (sets) in either a discrete
or a continuous sample space.
1. Probability of a union



Let A and B be events on a sample space Ω.
There are two cases for the probability of the
union A∪B:
1. A and B are mutually exclusive (no overlap):
i.e. A ∩ B = ∅.
2. A and B are not mutually exclusive: A ∩ B =
∅.
Explanation
2. Probability of an intersection



There is no easy formula for P(A ∩ B).
We might be able to use statistical independence
(Section 1.16).
If A and B are not statistically independent, we
often use conditional probability (Section 1.10.)
3. Probability of a complement
1.2.4 The Partition Theorem
The Partition Theorem.
1.2.5 Examples of basic probability
calculations




300 Australians were asked about their car
preferences in 1998. Of the respondents, 33% had
children. The respondents were asked what sort of
car they would like if they could choose any car at
all. 13% of respondents had children and chose a
large car. 12% of respondents did not have children
and chose a large car.
Find the probability that a randomly chosen
respondent:
(a) would choose a large car;
(b) either has children or would choose a large car
(or both).
First formulate events:
Respondents were also asked their opinions on car reliability
and fuel consumption. 84% of respondents considered
reliability to be of high importance, while 40% considered fuel
consumption to be of high importance.
Formulate events:
R = “considers reliability of high importance”,
F = “considers fuel consumption of high importance”.
Probability that respondent considers BOTH reliability AND
fuel consumption of high importance.

(f) Find the probability that a respondent
considered reliability, but not fuel consumption,
of high importance.
1.2.6 Formal probability proofs:
nonexaminable
i)
§1.3 Conditional probability
1.3.1 Conditional Probability


Conditioning is another of the fundamental tools
of probability: probably the most fundamental
tool. It is especially helpful for calculating the
probabilities of intersections, such as P(A∩B),
which themselves are critical for the useful
Partition Theorem.
Additionally, the whole field of stochastic
processes is based on the idea of conditional
probability. What happens next in a process
depends, or is conditional, on what has happened
beforehand.
Dependent events

Suppose A and B are two events on the same
sample space. There will often be dependence
between A and B. This means that if we know
that B has occurred, it changes our knowledge of
the chance that A will occur.
Example: Toss a dice once.
Conditioning as reducing the sample space
P( A | B) 
2 2 / 36 P( A  B)


3 3 / 36
P( B)
Definition of conditional probability
Conditional probability provides us with a way to reason about
the outcome of an experiment based on partial about the
outcome of an experiment, based on partial information.
Conditional Probabilities Satisfy the Three
Axioms
Conditional Probabilities Satisfy General
Probability Laws
Simple Example using Conditional
Probabilities
The Multiplication Rule
1.3.2 Multiplication (Chain) Rule:
Example
Example Three cards are drawn from an ordinary 52card deck without replacement (drawn cards are not
placed back in the deck). We wish to find the
probability that none of the three cards is a “heart”.
1.3.3 Total Probability Theorem
Example Using Total Probability Theorem
You enter a chess tournament where your probability of
winning a game is 0.3 against half the players(call them type
1),0.4 against a quarter of the player (call them type 2), You
play a game against a randomly chosen opponent. What is the
probability of winning?
1.3.4 Bayes’ Theorem: inverting
conditional probabilities
Example The False-Positive Puzzle.
§1.4 Independence of the events
1.4.1 Statistical Independence
We can extend the definition
to arbitrarily many events:
Statistical independence for calculating the
probability of an intersection


In section 1.3 we said that it is often hard to calculate P(A∩
B).
We usually have two choices.
Pairwise independence does not imply
mutual independence