THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1306 INTRODUCTORY STATISTICS
(SEMESTER 1 2008/2009)
2
Probability
To most people, probability is a loosely de°ned term employed in everyday±s
conversation to indicate the measure of one±s belief in the occurrence of a future
event. Say, what is the chance of arriving on time if one takes the bus? takes the
taxi?
However, for scienti°c purposes, it is necessary to give the word probability a
de°nitive, clear interpretation.
One can think of probability as the language in
discussing uncertainty.
For instance, in tossing a coin (random experiment), the
outcomes can either be
f
H
g
or
f
T
g
. If we know exactly what the outcome of the
next trial will be, then we are certain that, say, a
f
H
g
will show up next. However,
it is impossible to know exactly what is going to happen in reality  uncertainty.
De°nition.
A
random experiment
is a process leading to at least two possible
outcomes with uncertainty as to which will occur. Examples are (i) tossing a coin;
(ii) rolling a dice; (iii) daily changes in Hang Seng Index.
De°nition.
The possible outcomes of a random experiment are called the
basic
outcomes
, and the set of all basic outcomes is called the
sample space
,
S
. Examples
are (i)
S
=
f
H; T
g
; (ii)
S
=
f
1
;
2
;
3
;
4
;
5
;
6
g
; and (iii)
S
=
f
up, down
g
.
De°nition.
An
event
is a set of basic outcomes, or a collection of some basic
outcomes from the sample space, and it is said to occur if the random experiment
gives rise to one of its constituent basic outcomes.
Example 2.1.
In a random experiment of throwing two dices, the interest falls
in the total of the numbers turned up by the two dices.
Therefore, the basic
outcomes are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and the sample space is
S
=
f
2
;
3
;
4
;
5
;
6
;
7
;
8
;
9
;
10
;
11
;
12
g
.
Let
A
,
B
and
C
be 3 events where
A
=
f
even
g
=
f
2
;
4
;
6
;
8
;
10
;
12
g
;
B
=
f
odd
g
=
f
3
;
5
;
7
;
9
;
11
g
; and
C
=
f
sum
= 8
g
=
f
(2
;
6)
;
(3
;
5)
;
(4
;
4)
;
(5
;
3)
;
(6
;
2)
g
.
De°nition.
The
probability
P
(
E
)
of an event
E
2
S
is a set function de°ned on a
class of all events satisfying the axioms of probability which will be discussed later.
Theorem 2.1.
If an experiment is repeated for
n
times, and if the outcomes do not
a/ect each other. Let
r
be the number of occurrences of event
A
. Then,
P
(
A
) =
r=n
as
n
! 1
where
P
(
A
)
is the longrun frequency.
1
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From the frequentist±s view of probability, the frequency interpretation of prob
ability is a longrun average.
More precisely, if the random experiment can be
repeated over and over again with
n
= total number of trials and
r
= number of
trials in which event
A
actually happened out of the
n
trials. Then,
P
(
A
)
is simply
the proportion or rate of occurrences of event
A
in the long run; or
P
(
A
) =
r
n
=
relative frequency of event
A
.
For example,
P
(new born baby is a male) =
0
:
53
.
Another example is given to
the coin tossing problem. Suppose a fair coin is tossed for
n
times and the event
E
=
f
H
g
. Then, the probability for the occurrence of event
E
is
P
(
E
) =
No. of heads
No. of experiments
.
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 Fall '08
 Prof
 Statistics, Conditional Probability, Probability, Probability theory

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