Hi,
Can someone tell me if P(A|B), where A and B are mutually exclusive events is '0' or 'A'
Can someone tell me if P(A|B), where A and B are mutually exclusive events is '0' or 'A'
(I am going to get really technical now; this is probably not helpful for anyone trying to study this material for the first time)Hi @JStir4521 Probability must be between zero and 100% inclusive such that either P(B) = zero or P(B) = 100% are allowed. If P(B) = 0%, then A and B must be be independent per the definition. That's the easier question, for me.
To be honest, I struggle with the question of mutual exclusivity because it is still possible that P(A) = 0%, also. I asked GPT and I notice the first reply is what you might expect but then when I offer the challenge it tells "Therefore, based on the given information, we cannot determine whether events A and B are mutually exclusive or not."
See https://chat.openai.com/share/f417d760-6c3b-43a0-97dd-4fabfd9b754e
... in this way, I continue to suspect the answer might be indeterminate ("cannot be determined") without more information about P(A). Interesting question!
P.S. Actually the definition I fed it is not the condition for mutually exclusive (it applies to either, duh), but I'm still unclear. Strictly speaking P(A and B) = zero satisfies mutually exclusive but I am NOT 100% confident this test applies when P(A) = P(B) = 0%.
Mutually exclusive and independent not necessarily mean the same thing:(I am going to get really technical now; this is probably not helpful for anyone trying to study this material for the first time)
The point is that P(A and B) = 0 is not the definition of mutual exclusivity. Mutual exclusivity of two event means that their intersection is the empty set, i.e. A AND B = EmptySet. Thus, P(A AND B) = P(EmptySet) = 0 (by definition of a probability measure).
Something that really bothered me as a mathematician in both reading through the GARP Book as well as watching your videos is the mantra (and even exercise question in the official book!) that "Mutually exclusive event cannot be independent". This is not only wrong sometimes - as shown by the example with one of the probabilities being zero; it is TRIVIALLY wrong in every conceivable probability space!
This is because by definition every probability space has to contain the empty set. Obviously, EmptySet AND EmptySet = EmptySet, i.e. the EmptySet is mutuially exclusive from itself. However:
P(EmptySet AND EmptySet) = P(EmptySet) = 0 = 0 * 0 = P(EmptySet) * P(EmptySet)
and hence the EmptySet is also independet from itself.
That means that every probability space by definition contains events that are both mutually exclusive and independent.
Just to be clear, P(A AND B) = 0 does not imply mutual exclusivity.Mutual exclusivity is the condition that the intersection of two events results in an empty set (A AND B = EmptySet). This is satisfied when P(A AND B) = P(EmptySet) = 0.