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1-1 Properties of Probability
(機率的性質)
● 建立時間 @2025年10月7日 下午12:40
☉ 課程
機率論
1-1 Properties of Probability ()
Definition (定義):
•
Probability measures the likelihood of an event occurring, expressed as a
value between 0 and 1.
。 P(E) is the probability of an event E.
。 0 ≤ P(E) ≤ 1
。 P(S) = 1, where S is the sample space (the set of all possible
outcomes).
1. Non-negativity ()
The probability of any event is always non-negative.
2. Normalization (1)
P(E) > OVE
The probability of the entire sample space S is equal to 1.
P(S) = 1
This means that, in any probability model, at least one of the possible outcomes
must occur.
3. Additivity (加法性)
For any two mutually exclusive (disjoint) events (A) and (B), the probability of
their union is the sum of their individual probabilities.
P(AUB) = P(A) + P(B), IFAN B = 0
This property generalizes to a countable number of disjoint events:
1-1 Properties of Probability (機率的性質)

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P(A1 A2 A3 U ... U An) :
-
P(A1) + P(A2) + P(A3) + ...
+
P(An)
where (A1,A2,....., An) are mutually exclusive events.
4. Complementary Rule (U)
The probability of the complement of an event (A) (denoted (A^c )) is given
by:
P(Ac)
=
1 - P(A)
This tells us the probability of an event not occurring is equal to 1 minus the
probability of the event occurring.
5. Sub-additivity (7)
For any events (A) and (B), the probability of their union is at most the sum of
their individual probabilities.
P(AUB) ≤ P(A) + P(B)
This is an important property used in more advanced probability theories, such
as measure theory.
6. Monotonicity ()
If event A is a subset of event (B) (i.e., (ASB)), then:
P(A) ≤ P(B)
This is because the probability of a smaller event cannot exceed that of a larger
event.
7. Independence ()
Two events (A) and (B) are independent if:
P(ANB) = P(A) * P(B)
This is a key concept in probability theory, especially in contexts like random
variables and stochastic processes.
8. Probability of a Union (NE)
For any two events (A) and (B), the probability of their union is given by:
P(AUB) = P(A) + P(B) − P(ANB)
1-1 Properties of Probability (*)
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