Claim:Conditional probabilities satisfy Kolmogorov's axioms

From CS2800 wiki
If [math]A [/math] is an event, then the function [math]Pr' : A \href{/cs2800/wiki/index.php/%5Cto}{\to} \href{/cs2800/wiki/index.php?title=Real_number&action=edit&redlink=1}{\mathbb{R}} [/math] given by [math]Pr'(B) := Pr(B|A) [/math] is a probability measure.

Informally, this means that conditional probabilities follow the same rules as normal probabilities. For example, you could use this to prove that [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(A \href{/cs2800/wiki/index.php/%5Csetminus}{\setminus} B \href{/cs2800/wiki/index.php/%5Cmid}{\mid} A) = 1 - \href{/cs2800/wiki/index.php/Pr}{Pr}(B \href{/cs2800/wiki/index.php/%5Cmid}{\mid} A) [/math] (using the fact that Pr(S ∖ E) = 1 - Pr(E)).