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Chapter I Random events and probabilities

§1.1 Random events and sample spaces

The task of probability theory is to find out the possibility of random phenomena, and to give a way to measure the probability and its algorithm.

A random trial is an observation of random phenomena.

① can be repeated under the same conditions.

② may have different results in each test, and it is not certain which result will occur in the end.

③ knew all the possible results of the experiment in advance.

Every possible result of a randomized trial becomes a random event, or event for short.

Events are divided into basic events and compound events. It can be divided into inevitable events (recorded as Ω) and impossible events (recorded as)

Sample space: the set of all the basic events generated by a random trial E is called the sample space (Ω), and the element is called a sample point.

Write it down as omega. Ω = {ω}.

§1.2 the relationship and operation of events

The inclusion of ① events is equal to

Sum (union) and product (intersection) of ② events

③ incompatible events and opposing events

Let An and B be two events. If An and B cannot occur at the same time, that is, AB=, calls An and B incompatible or mutually exclusive events.

If An and B are incompatible and their sum is an inevitable event, that is, AB= and A ∪ B = Ω, then An and B are called opposing events or inverse events.

The difference between two events in ④

Let An and B be two events, "event An occurs but event B does not occur" is an event, called the difference between events An and B, recorded as Amurb.

The operational nature of the event:

① commutative law: a ∪ Baub ∪ Aje ABBALB

② binding law: a ∪ (B ∪ C) = (A ∪ B) ∪ C, (AB) Centra (BC)

③ distribution law: a (B ∪ C) = (AB) ∪ (AC)

A ∪ (BC) = (A ∪ B) (A ∪ C)

④ de Morgan (De Morgan) duality law:

The inverse of (A ∪ B) = the inverse of An and the inverse of B; the inverse of AB = the inverse of A ∪ B.

§1.3 probability of events and its calculation

Statistical definition of probability-- definition 1.1: P (A) ≈ nadir N

① nonnegative ② canonical ③ finite additivity

Classical probability: ① finiteness ② and other possibilities

P (A) = (all sample points in A) / total number of sample points in Ω = mporium

Counting rules of n: addition principle, multiplication principle

Hypergeometric distribution

Geometric probability

§1.4 axiomatic definition of probability

The sample space of the random trial E is Ω, and the set family of all events of E, including Ω, is £, if any of the events An of £

Can give a real number P (A), and P (A) satisfies the condition:

If ① is non-negative: 00, then P (Ai | B) = P (B | Ai) P (Ai) / ((juni1Lingn) ∑ P (B | Aj) P (Aj))

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