1
Chapters 4 Probability
Theory – Study of
Randomness

2
Idea of
probability
A phenomenon is
random
if
individual outcomes are
uncertain but there is a
regular distribution of
outcomes in a large number
of repetitions.
Probability of an outcome of
a random phenomenon is
the proportion of time it
would occur in a very long
series of repetitions.

3
Basic definitions
Sample space --
the set of
all possible outcomes of a
random phenomenon.
Event
a subset of the
sample space
Probability model
Description of sample
space
A way to assign
probabilities to events

4
Assigning probabilities
to sample spaces
Assign a probability to each
individual outcome.
Each
probability is a number
between 0 and 1 and
the
probabilities must sum to 1.
The
probability of any
event
is the sum of the
probabilities of the
outcomes making up the
event.

5
Basic Probability Rules
and Definitions
The probability of any event
is between 0 and 1.
If S is the sample space,
then P(S) =1.
The
complement
of an
event A is the event, A
c
, that
A doesn’t occur.
P(A
c
) =
1- P(A)

6
Basic Probability
Rules and
Definitions
Two events, A and
B,
are
disjoint
if they
have no elements
in
common and,
therefore, can’t occur
simultaneously.
We
then have
P(A or B)
= P(A or B occurs) =P(A) + P(B)

7
Addition rule for
several disjoint events
If A, B, and C are disjoint
events then
P(at least one of A, B, C )
= P(A) +P(B) + P(C
).
This extends to any number of
disjoint events.

8
Independent
Events
Two events, A and B, are
independent
if knowing that
one occurs does not change
the probability that the other
occurs.
If A and B are independent,
then
P(A and B) = P(A)P(B)
This is called the
multiplication rule for
independent events.