Independence and Incompatibility of Random Events

In probability theory, independence and incompatibility are concepts related to the relationship between random events.

1. Independence: Two events are considered independent if the occurrence or non-occurrence of one event does not affect the probability of the occurrence or non-occurrence of the other event. In other words, knowing whether one event happens provides no information about the likelihood of the other event happening.
   Events A and B are independent if P(A intersection B) = P(A)*P(B);
2. Incompatibility: Two events are incompatible if they cannot occur simultaneously. If the occurrence of one event excludes the possibility of the other event happening, they are considered incompatible. For example, flipping a coin and getting heads and tails simultaneously is incompatible since the coin can only show one side at a time.
   Events A and B are incompatible if P(A intersection B) = 0.

Examples of independent and incompatible events: