What are the relations and differences between incompatibility, opposition, independence and irrelevance in probability theory?

1. Differences and connections between incompatible and opposing events.

Incompatibility: one will happen and the other will not happen. They are in a probability space.

Contradictory events: One of the two mutually exclusive events must occur.

Difference and connection: two events may or may not occur in mutual incompatibility, and one and only one event occurs in the opposite event.

2. The connection and difference between independence and irrelevance.

Independent event: two events are not related to each other, or they can refer to different probability events, which are not in a phase probability space.

Irrelevant events: nonlinear correlation.

Difference and connection: independence must be irrelevant, and irrelevant is not necessarily independent.

Extended data:

Two random variables x, y, two events a and b.

Independence of A and B means that P(A∩B)=P(A)*P(B), and independence of X and Y means that any event A defined by X is independent of any event B defined by Y.

A and b are incompatible with each other, which means that P(A∩B)=0.

X and y are not related to each other, that is, they are not linearly related (covariance Cov(X, y) is zero), but they are not necessarily independent.

References:

Baidu Encyclopedia-Opposite Event Baidu Encyclopedia-mutually exclusive events Baidu Encyclopedia-Independent of each other