Conditional Probability

Abstract

Let $(\varOmega,\mathcal{F},\mathrm{P})$ be a probability space and $A \in \mathcal{ F}$ an event having strictly positive probability. Recall that the conditional probability of P with respect to A is the probability P _A on $(\varOmega,\mathcal{F})$, which is defined as

$$\displaystyle{\mathrm{P}_{A}(B) ={ \mathrm{P}(A \cap B) \over \mathrm{P}(A)} \quad \text{for every }B \in \mathcal{ F}\ .}$$

Intuitively the situation is the following: initially we know that every event $B \in \mathcal{ F}$ can appear with probability P(B). If, later, we acquire the information that the event A has occurred or will certainly occur, we replace the law P with P _A , in order to keep into account the new information.