AMS 311,
Lecture 5, Spring Semester, 2010
2.4
Some Simple Propositions
Proposition 4.1. For any event
E
,
)
(
1
)
(
E
P
E
P
C

=
.
Proposition 4.2. If
F
E
⊆
, then
)
(
)
(
F
P
E
P
≤
.
Also, if
F
E
⊆
, then
)
(
)
(
)
(
)
(
E
P
F
P
FE
P
E
F
P
C

=
=

.
Proposition 4.3.
If
E
and
F
are any two events
,
)
(
)
(
)
(
)
(
EF
P
F
P
E
P
F
E
P

+
=
∪
.
Boole’s (Bonferroni’s) Inequality
If
E
and
F
are any two events
,
)
(
)
(
)
(
F
P
E
P
F
E
P
+
≤
∪
.
This is a crucial fact to have handy in applied statistical work.
If
E
and
F
are any two events
,
)
(
)
(
)
(
C
EF
P
EF
P
E
P
+
=
.
Odds
in favor of an event
A
are
r
to
s
if
s
r
r
A
P
+
=
)
(
. Odds against an event
A
are
r
to
s
,
if
s
r
s
A
P
+
=
)
(
. If
p
A
P
=
)
(
,
then the odds in favor of
A
are
p
to
p

1
.
Generalizations of probability of union of two events:
If
3
2
1
,
,
E
E
E
are any three events, then
).
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
3
2
1
3
2
3
1
2
1
3
2
1
3
2
1
E
E
E
P
E
E
P
E
E
P
E
E
P
E
P
E
P
E
P
E
E
E
P
+



+
+
=
Example. A doctor has 520 patients, of which
1.
230 are hypertensive (T)
2.
185 are diabetic (D)
3.
35 are hypochondriac (C) and diabetic
4.
25 are all three
5.
150 are none
6.
140 are only hypertensive
7.
15 are hypertensive and hypochondriac but not diabetic.
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 Spring '08
 Tucker,A
 Probability theory, e. Example, Simple Propositions Proposition

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