Probability

1) If A and B are mutually exclusive, then A n B is zero, by definition. Whereas independent means A occurring does not affect the probability B occurring. If they are M.E. then if A happens B cannot (no matter what the probabilities), and the fact AnB is empty suggests..?

2) A is a member of B, is kind of like saying All A's are B's, but not all B's are A's. On a Venn diagram, the A circle would be entirely inside the B circle.

3) .. is there something wrong with just saying no? or do you mean you're not sure how to justify your answer?

1. If they're independent, P(AnB)=...
2. If they are independent, P(B|A)=...
3. You can just calculate the probability at 50 tries.

3) Do you mean (0.98) ^50 ? or...

1) So YES they can occur ?

3) I mean i am not sure how to justify my answer. Do i have to write an answer saying why it is wrong, or can i just use maths?

1) No they can't. AnB is zero, which means A and B are mutually exclusive. Since being mutually exclusive means if A occurs, B can't and vice versa, they clearly affect each other and therefore cannot be independent.

3) to be honest, neither am I. But you are right; it's not a valid argument.

right thanks.

2) So that means A is SMALLER than B... And because Independent means "A occurring does not affect the probability B occurring", it means they cant exist together because they are within the same circle?

God I hate probability...

1) No they can't. AnB is zero, which means A and B are mutually exclusive. Since being mutually exclusive means if A occurs, B can't and vice versa, they clearly affect each other and therefore cannot be independent.

3) to be honest, neither am I. But you are right; it's not a valid argument.

Haha! Well yeah; If A occurs, then by the fact A is a subset of B, B occurred. At the end of it all, B is independent of A, but A is not independent of B (it literally depends on whether B has happened if A can even happen at all).

So to the original question.. no, they cannot be independent of each other.

If B is empty, P(AnB)=0=P(A)*P(B), so A, B independent.

3) Let A be the whole set. If B is a subset of A, we find P(AnB)=P(B)=P(B)*P(A), thus independent.

This is only true if A or B have 0 probability. In general, for A and B which have non-zero probability, A and B are either mutually exclusive or independent; they cannot be both. Since AnB is zero, this implies mutually exclusive, which implies not independent.

I'm a little lost with this.. A is supposed to be a subset of B, and since A is entirely contained by B, then P(AnB)=P(A). However if A and B are non-zero, and not 1, then P(AnB)=P(A)P(B) implies P(A)=P(A)P(B) which cannot hold, and thus cannot be independent. More intuitively, because A is a subset of B, that makes it a necessary requirement that B has to occur to let A occur, and that means they are not independant of eachother; A needs B to happen.