P(A∩B∩C∩D) stands for the probability of the simultaneous occurrence of A.B.C.D4 events, which is a generalization of the probability multiplication formula P(AB)=P(A)×P(B|A).

The error-prone point of probability multiplication formula is that it is easy to write P (AB) = P (A) × P (B), P (ABCD) = P (A) P (B) P (C) P (D).

Supplementary explanation:

1. addition rule

Theorem: Let A and B be mutually incompatible events (AB=φ), then:

P(A∪B)=P(A)+P(B)-P(AB)

Inference 1: If A 1, A2, …, An are incompatible with each other, then: P (A1+A2+...+An) = P (A1)+P (A2)+…+P (An).

Inference 2: Let A 1, A2, ..., A form a complete event group, then: P (A1+A2+...+An) =1.

Inference 3:? Contrary to event a.

Inference 4: if b contains a, then P (B-A) = P (B)-P (A).

Inference 5 (generalized addition formula):

For any two events a and b, there is P (A ∪ B) = P (A)+P (B)-P (AB).

2. Conditional probability:

The probability that A appears under the condition that event B is known is called conditional probability, which is recorded as: P(A|B).

Conditional probability calculation formula:

When P(A) >; 0，P(B|A)=P(AB)/P(A)

When P(B)>0, P(A|B)=P(AB)/P(B)

3. Multiplication formula:

P(AB)=P(A)×P(B|A)=P(B)×P(A|B)

Promotion: P(ABC)=P(A)P(B|A)P(C|AB)

4. Total probability formula

Suppose: If events A 1, A2, …, An are incompatible with each other, and A 1+A2+…+an = ω, then A 1, A2, …, an constitutes a complete event group.

The form of total probability formula is as follows:?