Probability of mutually exclusive event

[y2alk]

Hi,

Can someone tell me if P(A|B), where A and B are mutually exclusive events is '0' or 'A'

 

[brian.field]

0

P(A|B) = P(A and B) / P(B)
P(A and B) = 0 if mutually exclusive

 

[y2alk]

Thanks @brian.field

 

[Amierul]

Hi @David Harper CFA FRM, I have a question here.
are mutually exclusive events, say A and B considered as independent events?

Thanks in advance.

 

[David Harper CFA FRM]

Hi @Amierul Mutually exclusive events cannot be independent, they must be dependent. Our independence rule is that P(A∩B) = P(A)*P(B), but when the events are mutually exclusive the probability that both occur is zero, P(A∩B) = 0. I'm a big fan of the contrapositive (see https://en.wikipedia.org/wiki/Contraposition) because always gives you a way to finding a second true statement. So we know that:

• If mutually exclusive → dependent; (aka, not independent). If this is P → Q, the contrapositive is not Q → not P such that our TRUE contrapositive is:
• if independent → not mutually exclusive

So, I didn't want to glibly answer in the following way, but hopefully you can see how my glib answer would be "mutually exclusive events are extremely dependent! if one, not the other and vice-versa is very dependent!" I hope that's helpful.

 

[Amierul]

Thanks @David Harper CFA FRM. I really like your contraposition.

 

[JStir4521]

Hi all. I had a further question here if possible. It is possible for an event to have zero probability (might be trivial, but allowed nonetheless). In this case, the definitions for mutually exclusive and independence hold. ie: assume event B had 0 probability. P(A∩B) = P(A)P(B) = P(A) * 0 = 0 = P(A∩B).

I appreciate that this gives no information and is trivial, but I don't see anywhere that it is a requirement for an event to have non-zero probability, and so this is allowed? Looking forward to being proved wrong

 

[David Harper CFA FRM]

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%.

 

[Dominik Hosefelder]

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%.

(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.

 

[gsarm1987]

(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.

Mutually exclusive and independent not necessarily mean the same thing:
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.

Although P(A AND B) = 0 means mutual exclusivity, this alone doesn't guarantee independence. The claim that mutually exclusive events cannot be independent is incorrect. For example, in every probability space, the empty set (EmptySet) is both mutually exclusive and independent from itself.

 

[Dominik Hosefelder]

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.

Just to be clear, P(A AND B) = 0 does not imply mutual exclusivity.
An easy counterexample is the intervall [0,1] with the uniform distribution. Let A = [0,1] and B = {rational numbers in [0,1]}. A well known fact is that P(B) = 0. Hence, P( A AND B ) <= P(B) = 0.
On the other hand, A AND B = B != EmptySet.

In particular we find that P(A AND B) = P(A) * P(B), i.e. A and B are independent.