How to deal with a probability conditioned on empty

[sabbagh80]

Hi everyone,
The question:

Pr( A|\Phi )=? where \Phi means empty
Is it equal to 1 or not?
what is the meaning of this probability?

thanks.

 

[disregardthat]

A|empty set = empty set is an event in the empty "probability space", for which you can't assign a probability measure, so it isn't a probability space after all. So A|empty set isn't an event in any probability space, so Pr(A|empty set) doesn't make sense.

The axioms for a probability measure P on a sigma algebra X is that P(empty set) = 0, and P(X) = 1. But in this case X = empty set, so P(X) = 0, so such a measure cannot exist.

 

[Stephen Tashi]

A|so Pr(A|empty set) doesn't make sense.

However, in practical problems you may need to dance around that obstacle.
For example, if W is a uniform random variable on [0,1] then you might want to define a function
F(z,x,y) = P(W > z | x < W and W < y)
This function wouldn't exist at values such as x = 4 and y = 3.
If you want to define F(z,x,y) there, you have to include an additional provision in its definition. The most common one would be to say:
F(z,x,y) = P(W > z | x < W and W < y) when the set defined by (x < W
and W < y) is not empty. F(z,x,y) = 0 otherwise.

This is not extending the theory of sample spaces. It is extending the definition of a function, rather like removing a discontinuity.

 

[disregardthat]

Sure, we can define what would otherwise be meaningless all day long. OP asked for the probability of A|empty set, and that cannot be interpreted in any other way than what the theory of probability of reference would yield for Pr(A|empty set). For that my answer is sufficient.

 

[Stephen Tashi]

sabbagh80,

Are you asking the question merely out of curiosity or did you have a specific application in mind that must deal with it?

 

[sabbagh80]

Stephen Tashi,
No, it is not related to an specific application. this question arose me when I was dealing with another problem and was not directly related to that problem.