Conditional Probability

  • Paolo Baldi
Part of the Universitext book series (UTX)


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.

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Paolo Baldi
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

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