In probability theory, what is the relation between mutually

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In probability theory, what is the relation between mutually exclusive events and independent events? Does it matter if we a talking about a single trial (for example toss a die once) or 2 trials (for example toss the same die twice)?

Events which are mutually exclusive cannot be independent and vice-versa. In independent events, the outcome of one event does not impact the other. In mutually exclusive events, the outcome of one event provides information about the outcome of another (specifically, by making the probability 0). In other words, for independent events:

p(A)|B = p(A) AND p(B)|A = p(B) (read as, the probability of event A given event B = the probability of event A: that is, the conditional probability is equal to the non-conditional probability)

For mutually exclusive events:

p(A)|B = 0

In a single roll of a die, the outcomes (1) and (2) are mutually exclusive. That is, if you roll a 1, you can't also roll a 2.

In two rolls of the tie, the outcomes are independent. Rolling a 1 the first time doesn't change your probability of rolling a 2 the second time.

To use another example, consider choosing a random person and asking about their gender and name.

The outcomes of their gender and last name are independent. Knowing that the person is male doesn't change the probability that their last name is Smith.

Assuming that no males are named "Mary" (in the real world that's a bad assumption, but it illustrates the point), the events (gender = male) and (first name = "Mary") are mutually exclusive. Both things can't be true at once.

Not totally sure what you mean by "Events which are mutually exclusive cannot be independent and vice-versa."

It's the "vice- versa" part that is bothering me.

I have been able to prove M implies not I and also I implies not M.

Your vice-versa part sounds like M impies not I and also not I imlies M.

Could you please clarify?

Also, how do we handle this situation?

Once again let's toss a single fair die(sample space 1,2,3,4,5,6) once. Let event A = roll a 7 and event B = roll an 8. Clearly P(A and B) =0 and P(A) x P(B) = 0. This seems to imply that A and B are both mutually exclusive and independent. I guess this is just an exception???