Feb 9’09
INTRODUCTION TO PROBABILITY
#4
The theory of probability provides the foundation for
statistical inference
.
BASIC TERMS
. In probability, an
experiment
is an activity or occurrence with an
observable result. Each repetition of an experiment is called a
trial
. The possible results
of each trial are called
outcomes
or
events
. The set of all possible outcomes for an
experiment is called the
sample space
for that experiment.
For example, a sample space
S
for the experiment of tossing a coin is made up of two
possible outcomes:
heads (
h
)
and
tails (
t
) . In the set notations, we may write
S = {h, t}
An
event
is a
subset
of a sample space
. Events are designated with capital letters:
A, B, …, E, …
which may have subscripts. Thus, in the experiment of tossing a coin, one
of two
mutually exclusive
events may occur: the event
A = {h}
, which represents the
outcome “heads” and the event
B = {t}
, which represents the outcome “tails”. (
The
events are said to be
mutually exclusive
if no two of them can occur together
).
Different types of events may be associated with the same experiment, and an
experiment may have
more than one
sample space associated with it
.
Example
: for the experiment of rolling a single
fair
six-sided die, if we consider the
events like showing a single number on the top face, then we may introduce a sample
space (of size 6)
S
= {1, 2, 3, 4, 5, 6}
which includes six possible mutually exclusive
outcomes. But we may associate with this experiment also some other events:
a)
the die shows an
odd
number
:
there are three possible events of this type:
E
1a
= {1},
E
2a
= {3},
E
3a
= {5}, and the corresponding sample space (of size 3)
is
S
a
= {1, 3, 5};
b)
the die shows a number
greater than
2
:
there are four possible events of such a type:
E
1b
= {3},
E
2b
= {4},
E
3b
= {5},
E
4b
= {6}, and the corresponding sample space
(of size 4) is
S
b
= {3, 4, 5, 6};
c)
the die shows a
multiple of
3
:
there are two possible events of this type:
E
1c
= {3} and
E
2c
= {6}, and the sample space (of size 2) is
S
c
= {3, 6}.
A probability of an event