What is the meaning of | in probability theory?

[Square1]

What is the meaning of :|: ?

So how would one read the following for example...

set X = { u "member of" set A :|: u has property P}

 

[mathman]

:|: seems to mean that the expression to the right is a condition. I have not seen that form - usually it is just | alone.

 

[SteveL27]

What is the meaning of :|: ?

So how would one read the following for example...

set X = { u "member of" set A :|: u has property P}

Have never seen that notation before. Assuming it has the same meaning as the more usual


set X = { u "member of" set A | u has property P}

I would pronounce that "such that."

 

[Stephen Tashi]

In probability theory, the "|" has a special meaning, not completely captured by the condition "such that". It's usually spoken as "given".

Probabilities are assigned to subsets of some space of possible outcomes. If A and B are sets in such a space then $A \cap B$ and $A | B$ both refer to the set of elements in $A \cap B$, but the probability $P(A | B)$ tells us that probability is to be computed as if the elements of the "space of possible outcomes" are only those elements which are in B while $P(A \cup B)$ tells us that the "space of possible outcomes" is the original space of possible outcomes.

Hence the meaning of the notation P(X) is not as simple as "P(X) means the probability of the set (or event) X". When the "|" sign is used, the P(.) notation also tells something about what is to be considered the space of possible outcomes.

 

[blue_raver22]

In probability theory, the symbol '|' is often used to represent the conditional probability of an event A given an event B, denoted as P(A|B). This means the probability of event A occurring, given that event B has already occurred.

In the context of set theory, the symbol '|' is used to represent the "such that" or "such that it satisfies" relation. In the given example, set X is defined as the set of all elements 'u' that are members of set A and also have the property P. This can be read as "set X is equal to the set of all elements 'u' such that 'u' is a member of set A and 'u' has property P."

The symbol ':' is used in set theory to represent the "such that" relation. In the given example, the use of two colons (:|:) can be read as "such that it satisfies." So, the overall meaning of the expression is "set X is equal to the set of all elements 'u' such that 'u' is a member of set A and 'u' has property P."