# Conditional probability as a function

Sample space (S) is a set of all possible outcomes of a random experiment.

$S = \{x | x$is an outcome$\}$

Event (E) is a subset of sample space.

$E \subseteq S$

Probability is a function from a set (Event) to $[0,1]$

$p : E -> [0,1]$

My doubt is that whether conditional probability is also a form of probability, whose argument is a set? Is conditional probability a function?

If yes, what is the argument of the conditional probability function?

• Is a function of a pair of events. – Mauro ALLEGRANZA Jan 16 '18 at 14:17
• @MauroALLEGRANZA Is it true that A/B is a set? – hanugm Jan 16 '18 at 14:22
• NO; but $A$ and $B$ are sets. – Mauro ALLEGRANZA Jan 16 '18 at 14:27
• Then A/B is a non-mathematical terminology? otherwise, what is $A/B$? Is it a number/ set/ .. ? Can we write A/B interms of sets A and B? – hanugm Jan 16 '18 at 14:32
• $P(A/B)$ is a symbol. We may write $P(A_B)$ or some other way... We define: "the conditional probability as the measure of the probability of an event $A$ given that (by assumption, presumption, assertion or evidence) another event $B$ has occurred." Thus, it is definied for a couple of events. – Mauro ALLEGRANZA Jan 16 '18 at 14:35

For an event $E$ s.t. $P(E)>0$, the answer is yes. Since $P(A|E)=\frac{P(A\cap E)}{P(E)}$ , we can define a new probability measure $P_E$ s.t. $P_E(A)=P(A|E)=\frac{P(A\cap E)}{P(E)}$. All the properties of a probability measure hold for $P_E$.
In some cases it is possible to define a conditional probability measure on events $E$ s.t. $P(E)=0$, but that is more complicated.
• In answer, you defined a new probability measure, which takes input a set $A$, but for actual conditional probability function? – hanugm Jan 16 '18 at 14:26
• It takes a set $A$ but instead of giving the original probability, it gives the probability under the condition that $E$ happened. – joeyg Jan 16 '18 at 14:30
• Just for clarification, $P(A|E)$ means the probability that $A$ occurs given that $E$ occured. – joeyg Jan 16 '18 at 14:43