1
Lecture Plan
•
Experiments, Outcomes and Events
•
The Axioms of Probability
•
Axiom consequences
•
Finite Sample Space.
2
Experiments, Outcomes, Sample Space, and Events
Example:
•
Experiment: Toss a coin three times.
•
Outcomes: The possible outcomes are
hhh, hht, hth, htt, thh, tht, tth, ttt
.
•
Sample space: The set of all outcomes:
S
=
{
hhh, hht, hth, htt, thh, tht, tth, ttt
}
•
Events: Are subsets of
S
to which we will assign probabilities.
Examples of events:
A
at least one head:
A
=
{
hhh, hht, hth, htt, thh, tht, tth
}
,
B
the first two tosses are
heads:
B
=
{
tth, ttt
}
An event
A
is said to have occurred if any outcome
ω
∈
A
occurs when an experiment is conducted.
Suppose an experiment is done with outcome
ω
=
tth
then all events (subsets of
S
) containing
ω
occur.
2.1
The Axioms of Probability
See section 2.4 pages 6974.
In probability we are interested in assigning numbers in [0
,
1] to certain subsets of
S
(called events). The
following axioms allow us to do this in a consistent way.
A1 If
A
⊂
S
is an event then
P
(
A
)
≥
0
.
A2
P
(
S
) = 1
A3 If
A
1
, A
2
, . . .
are mutually exclusive events then
P
(
∪
n
i
=1
A
i
) =
n
X
i
=1
P
(
A
i
)
for all
n
= 1
,
2
, . . . ,
∞
.
1
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Note: In order for a set of outcomes to be an event we need to be able to assign a probability to the set.
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 Spring '04
 GALLEGO
 Probability, Probability theory, Coin flipping, Coin

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