Stochastic Calculus pp 85-107 | Cite as

# Conditional Probability

Chapter

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## 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 Intuitively the situation is the following: initially we know that every event \(B \in \mathcal{ F}\) can appear with probability P(

*A*is the probability P_{ A }on \((\varOmega,\mathcal{F})\), which is defined as$$\displaystyle{\mathrm{P}_{A}(B) ={ \mathrm{P}(A \cap B) \over \mathrm{P}(A)} \quad \text{for every }B \in \mathcal{ F}\ .}$$

*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.## Copyright information

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