What are the Different Types of Events in Probability Theory?

[zorro]

I never understand the difference between 'Equally Likely Events', 'Mutually Exclusive Events' and 'Exhaustive events'. I also get confused in calling an event to be of more than one type.

I would be grateful if someone explains it clearly giving examples.

 

[CompuChip]

Equally likely means, that the probability for each event is the same. For example, if you throw a coin, probability for either side is 50%. If you throw a die, the probability for any outcome is 1/6.

Mutually exclusive means, that if one event occurs, the other cannot. For example, throwing a die, "the number is even" and "the number is odd" are mutually exclusive events: if it is even it is not odd, and if it is odd it is not even. "The number is 5 or 6" and "the number is less than 5" are also mutually exclusive. "The number is 5 or 6" and "the number is even" are not mutually exclusive, because if it is 6, then it is both.

Exhaustive events means that the events cover all possibilities.
For example "the number is 1", "the number is 2", "the number is 3", "the number is 4", "the number is 5" and "the number is 6" is a list of exhaustive events for throwing a die: if you throw a die, you know that one of them will occur.

In fact the events of the last example are equally likely (the probability for all of them is the same, namely 1/6), they are mutually exclusive (if one of them happens, neither of the other ones happens) and they are exhaustive (at least one of them will happen).

 

[zorro]

Thanks a lot CompuChip. You explained it in a very simple language.

 

[zorro]

@CompuChip

I forgot to mention one more question in O.P.
What is the basic difference between independent and mutually exclusive events? Can they be equal in any case?

 

[CompuChip]

Independent means, that the probability of one event does not depend on the other.
For example, if you throw two dice the probability of gettings a 6 on either of them is independent of the probability of getting a 6 on another. So the event "getting 6 on the first die" and the event "getting heads on the second die" are independent.
To contract, the events "getting 6 on the first die" and "throwing a total of 10" are not independent, because the probability of the second event happening depends on whether the first one happens or not.

 

[zorro]

To contract, the events "getting 6 on the first die" and "throwing a total of 10" are not independent, because the probability of the second event happening depends on whether the first one happens or not.

Is it that in this case they are dependent and non-mutually exclusive events?
Is there any case where two events are independent as well as mutually exclusive?

 

[Bacle]

Think about it AQ:

P(A/\B)=P(A)P(B) , if/when A,B are independent. What if A,B are mutually-exclusive?

 

[zorro]

If A and B are mutually exclusive then P(A/\B) = 0
So mutually exclusive events can never be independent except when probability of one of them is 0. Is it correct?

 

[Bacle]

Precisely, tho, at least in the discrete case, I believe if P(E)=0 for some event E,
then E would not be in the sample space.

 

[zorro]

A null event is a subset of every sample space, as far as I know.

 

[Bacle]

Well, this may be an issue of technicalities, but I don't know if a null event refers
to any subset with probability zero. I am not clear on how the null event is defined
in general. I imagine it may be defined as having no outcome in our experiment/observation,
but I am not sure.
I don't think we would, say, include the outcome of rolling a 7 when we roll a standard
die.

But yes, technically, you are correct, as the empty set is a subset of every set.
Still, there is a qualitative difference between events of measure zero in continuous
sample spaces , e.g., selecting a number at random in the unit interval vs. rolling
a die once.

Maybe a good illustration of disjointness vs. independence would be this: we have
a standard die, with even-numbered sides colored red, odd-numberscolored blue.

Then we roll the die : what is the probability that the die landed in the number 6,
if we know the die landed in a blue face?.

 

[Bacle]

Just wanted to follow up with something: an event may have probability zero
without being the empty set; I thought about it when reading another post.

Specifically, think of the case of throwing a dart to hit the real line. Then
the event of hitting a rational number (or, for that matter, the event of
hitting anyone specific number on the real line) is 0, but it is part of the
sample space.

Sorry to bother with this, but I thought it may help clarify some points; it did
so for me.