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Erschienen in:
2017 | OriginalPaper | Buchkapitel
4. Conditional Probability
verfasst von : Paolo Baldi
Erschienen in: Stochastic Calculus
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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 A is the probability P
A
on \((\varOmega,\mathcal{F})\), which is defined as 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.
$$\displaystyle{\mathrm{P}_{A}(B) ={ \mathrm{P}(A \cap B) \over \mathrm{P}(A)} \quad \text{for every }B \in \mathcal{ F}\ .}$$