The difference between stand-alone events and mutexclusive events is mainly reflected in the following two aspects:
Properties: A mutexclusive event is one in which the intersection of two events A and B is empty, i.e., it is impossible for the two events to occur at the same time. Independent events, on the other hand, do not affect each other, i.e., the occurrence of one event does not affect the occurrence of another.
Probability relation: If two events are independent of each other, then they should satisfy the equation p(ab)=p(a)p(b). There is no such probabilistic relationship for mutually exclusive events.
As for the connection between independent events and mutually exclusive events, both are basic concepts in probability theory, and they describe the relationship between two or more events. Note, however, that independent events and mutually exclusive events are not the same concept, and there is no necessary connection between them. That is, two events may be independent or mutually exclusive, or neither independent nor mutually exclusive.
There are many examples of stand-alone events and mutually exclusive events, here are some examples:
Coin toss: Heads and tails are mutually exclusive events because they cannot occur at the same time. At the same time, each coin toss is a separate event, as the outcome of each coin toss is not affected by previous coin tosses.
*: The probability of winning is the same for everyone, so the event of winning is independent for each person. At the same time, one person winning the lottery does not affect the other person winning the lottery, so they are also mutually exclusive.
Playing cards: Everyone's cards are randomly assigned, so each person's hand is independent. At the same time, the cards in one person's hand do not affect the cards in the other's hand, so they are also mutually exclusive.
Watching movies: Everyone's evaluation of a movie is independent, and one person's evaluation does not affect another's evaluation. At the same time, the evaluations of two people are not mutually exclusive, since they can occur at the same time (for example, two people can watch the same movie at the same time and give their own evaluations).
These examples illustrate the different nature of independent events, with independent events being the occurrence of one event that is not affected by another, and mutually exclusive events being when two events cannot occur at the same time.