18.05
Lecture
2
February
4,
2005
§
1.5
Properties
of
Probability.
1.
P
(
A
)
⊂
[0
,
1]
2.
P
(
S
)
=
1
3.
P
(
⇒
A
i
)
=
P
(
A
i
)
if
disjoint
↔
A
i
∞
A
j
=
≥
,
i
=
j
∈
The
probability
of
a
union
of
disjoint
events
is
the
sum
of
their
probabilities.
4.
P
(
≥
)
,
P
(
S
)
=
P
(
S
⇒≥
)
=
P
(
S
)
+
P
(
≥
)
=
1
where
S
and
≥
are
disjoint
by
de±nition,
P
(S)
=
1
by
#2.,
therefore,
P
(
≥
)
=
0.
5.
P
(
A
c
)
=
1
−
P
(
A
)
because
A,
A
c
are
disjoint,
P
(
A
A
c
)
=
P
(
S
)
=
1
=
P
(
A
)
+
P
(
A
c
)
⇒
the
sum
of
the
probabilities
of
an
event
and
its
complement
is
1.
6.
If
A
√
B,
P
(
A
)
←
P
(
B
)
by
de±nition,
B
=
A
⇒
(
B
\
A
)
,
two
disjoint
sets.
P
(
B
)
=
P
(
A
)
+
P
(
B
\
A
)
∼
P
(
A
)
7.
P
(
A
⇒
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
AB
)
must
subtract
out
intersection
because
it
would
be
counted
twice,
as
shown:
write
in
terms
of
disjoint
pieces
to
prove
it:
P
(
A
)
=
P
(
A
\
B
)
+
P
(
AB
)
P
(
B
)
=
P
(
B
\
A
)
+
P
(
AB
)
P
(
A
⇒
B
)
=
P
(
A
\
B
)
+
P
(
B
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 Summer '06
 DrStag
 Probability, Probability theory, different face values, disjoint Ai Aj

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