Show that the conditional probability $P(\cdot | C)$ satisfies the axioms of probability, and is thus, indeed, a probability function.

Question

The Statement of the Problem:

For a fixed event $C$ such that $P(C) \gt 0$, show that the conditional probability $P(\cdot | C)$ satisfies the axioms of probability, and is thus, indeed, a probability function.

Where I Am:

So, I'm stuck on this one mostly because I don't remember what this "dot" notation means, and can't seem to find anything anywhere on the web or in the text that explains it. Would someone be able to help me out here?

Answer 1

$P(A \mid C) =$ the probability of $A$ occurring, given that $C$ already occurred. Take a look at conditional probability. The notation $P(\cdot \mid C)$ refers to the map $A \mapsto P(A\mid C)$.