AMS 311, Lecture 6, Spring 2010
2.6.
Probability as a Continuous Set Function
A sequence of events
}
1
,
{
≥
n
E
n
is said to be an increasing sequence if
⊂
⊂
⊂
⊂
⊂
+
1
2
1
n
n
E
E
E
E
. If
}
1
,
{
≥
n
E
n
is an increasing sequence of events, then
we define
.
∞
=
∞
→
=
1
lim
i
i
n
n
E
E
.
A sequence of events
}
1
,
{
≥
n
E
n
is said to be an decreasing sequence if
⊃
⊃
⊃
⊃
⊃
+
1
2
1
n
n
E
E
E
E
. If
}
1
,
{
≥
n
E
n
is an decreasing sequence of events, then
we define
∞
=
∞
→
=
1
lim
i
i
n
n
E
E
.
Proposition 6.1. (Continuity of Probability Function) For any increasing or decreasing
sequence of events,
}
1
,
{
≥
n
E
n
,
)
lim
(
)
(
lim
n
n
n
n
E
P
E
P
∞
→
∞
→
=
.
2.7.
Probability as a Measure of Belief
A rational gambler would define subjective probabilities that satisfy all of the axioms of
probability. In that event, the theorems developed in this chapter are true.
Example. Suppose that in a 7horse race you feel that each of the first 2 horses has a 20
percent chance of winning, horses 3 and 4 each has a 15 percent change, and the
remaining 3 horses, a 10 percent chance each.
Would it be better for you to wager at
even money, that the winner will be one of the first three horses, or to wager, again at
even money, that the winner will be one of the horses 1, 5, 6, 7?
In practice, most people propose probabilities that do not satisfy the axioms of
probability. This happens when people use odds. Recall the definition of odds:
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
. The odds ratio
of an event
A
is defined by
.
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 Spring '08
 Tucker,A
 Conditional Probability, Probability, Probability theory

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