# How to deal with a probability conditioned on empty

• sabbagh80
In summary, Pr(A|empty set) doesn't make sense because an event that doesn't exist cannot be assigned a probability.
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.

Last edited:
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.

disregardthat said:
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.

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.

sabbagh80,

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

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.

## 1. What is a probability conditioned on empty?

A probability conditioned on empty is a probability calculation that takes into account the possibility of an outcome being empty or having no occurrence. This is often used in situations where there are multiple possible outcomes, but some of them may result in nothing happening.

## 2. How do you calculate a probability conditioned on empty?

The calculation for a probability conditioned on empty involves dividing the probability of the desired outcome by the total number of possible outcomes, including the possibility of the outcome being empty. This can be represented mathematically as P(A|B) = P(A∩B) / P(B+E), where A is the desired outcome, B is the condition, and E is the possibility of an empty outcome.

## 3. What is an example of a probability conditioned on empty?

An example of a probability conditioned on empty is the probability of rolling a sum of 7 on two fair dice when one of the dice is red and the other is blue. The desired outcome is a sum of 7 (A), the condition is one of the dice being red (B), and the possibility of an empty outcome is if the red die rolls a 1 and the blue die rolls a 6 (E).

## 4. How does the possibility of an empty outcome affect the overall probability?

The possibility of an empty outcome will decrease the overall probability, as it is considered a possible outcome that does not contribute to the desired outcome. This means that the probability of the desired outcome will be divided by a larger total number of outcomes, resulting in a smaller probability.

## 5. When should a probability conditioned on empty be used?

A probability conditioned on empty should be used when there is a possibility of an outcome being empty or having no occurrence, and this possibility needs to be taken into account in the calculation. This is often used in situations where there are multiple possible outcomes, but some of them may result in no impact or no change.

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