What is the probability formula of independent events occurring simultaneously? What is the probability addition formula of mutually exclusive time? What are these probability formulas?

What is the probability formula of independent events occurring simultaneously? What is the probability addition formula of mutually exclusive time? What are these probability formulas?

Mutual independence means that the occurrence of one event will not affect the occurrence probability of another event
That is, whether event a occurs or not, the probability of event B is the same, P (B / a) = P (b);
Similarly, whether B occurs or not, the probability of event a is the same, P (A / b) = P (a);
Usually, the probability of two things happening at the same time p (AB) = P (A / b) P (b) = P (B / a) P (a) (whether a and B are independent or not, this formula holds). To judge whether an event is independent or not is to follow the above definition, just like your classmate has a cold (event B), so the probability of your catching a cold with him is not equal to the probability of your catching a cold, Therefore, the probability of you catching a cold at the same time p (AB) = P (A / b) P (b); on the contrary, whether someone in Africa has a cold (event B) or not does not affect whether you have a cold (assuming there is no butterfly effect or something like this), then p (A / b) = P (a). Similarly, whether you have a cold or not does not affect him, P (B / a) = P (b)
Therefore, the probability of simultaneous occurrence of independent events a and B: P (AB) = P (a) P (b)
Mutex means that as long as event a occurs, event B cannot occur; otherwise, it also holds, that is, P (AB) = 0, because P (B / a) = P (A / b) = 0
The probability of mutually exclusive events a and B: P (a + b) = P (a) + P (b)
Omitting examples is not good at this~
Mutual independence and mutual exclusion are opposite concepts in a sense. If two events are mutually exclusive, they are definitely not independent. If they are mutually independent, they are definitely not mutually exclusive
That’s it,hope it is easy to understand~