p. 21
Sample Space and Events
•
Sample Space
: the set of all possible outcomes in a random
phenomenon. Examples:
1.Sex of a newborn child:
= {girl, boy}
2.The order of finish in a race among the 7 horses 1, 2, …, 7:
= {all 7! Permutations of (1, 2, 3, 4, 5, 6, 7)}
3.Flipping two coins:
= {(H, H), (H, T), (T, H), (T, T)}
4.Lifetime of a transistor:
= [0,
)
•
Event
: Any (measurable) subset of
is an event. Examples:
1.
A
={girl}: the event  child is a girl.
2.
A
={all outcomes in
starting with a 3}: the event  horse 3 wins
the race.
3.
A
={(H, H), (H, T)}: the event  head appears on the 1st coin.
4.
A
=[0, 5]: the event  transistor does not last longer than 5 hours.
•
an event occurs: outcome
the event
•
Q
: How many different events if #
=
n
<
?
p. 22
• Set Operations of Events
Union.
C
: either
A
or
B
occurs
Intersection.
C
: both
A
and
B
occur
Complement.
C
:
A
does not occur
Mutually Exclusive.
A
and
B
have no
outcomes in common.
Definitions of union and intersection for more than two
events can be defined in a similar manner
• Some Simple Rules of Set Operations
Commutative Laws.
Associative Laws.
Distributive Laws.
DeMorgan’s Laws.
A
∪
B
=
B
∪
A
and
A
∩
B
=
B
∩
A
(
∪
n
i
=1
A
i
)
c
=
∩
n
i
=1
A
c
i
and
(
∩
n
i
=1
A
i
)
c
=
∪
n
i
=1
A
c
i
.
A
∩
B
=
∅⇒
C
=
A
c
⇒
C
=
A
∩
B
⇒
C
=
A
∪
B
⇒
(
A
∪
B
)
∩
C
=(
A
∩
C
)
∪
(
B
∩
C
)
(
A
∩
B
)
∩
C
=
A
∩
(
B
∩
C
)
.
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 Fall '11
 ShaoWeiCheng
 Permutations, Probability, Probability theory, Empty set, Probability space, A1 A2

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