Conditional Probability - Formal Derivation

Formal Derivation

Formally, is defined as the probability of according to a new probability function on the sample space, such that outcomes not in have probability 0 and that it is consistent with all original probability measures.

Let be a sample space with elementary events . Suppose we are told the event has occurred. A new probability distribution (denoted by the conditional notation) is to be assigned on to reflect this. For events in, it is reasonable to assume that the relative magnitudes of the probabilities will be preserved. For some constant scale factor, the new distribution will therefore satisfy:

Substituting 1 and 2 into 3 to select :

$\begin{align} \sum_{\omega \in \Omega} {P(\omega | B)} &= \sum_{\omega \in B} {\alpha P(\omega)} + \cancelto{0}{\sum_{\omega \notin B} 0} \\ &= \alpha \sum_{\omega \in B} {P(\omega)} \\ &= \alpha \cdot P(B) \\ \end{align}$

So the new probability distribution is

Now for a general event ,

$\begin{align} P(A|B) &= \sum_{\omega \in A \cap B} {P(\omega | B)} + \cancelto{0}{\sum_{\omega \in A \cap B^c} P(\omega|B)} \\ &= \sum_{\omega \in A \cap B} {\frac{P(\omega)}{P(B)}} \\ &= \frac{P(A \cap B)}{P(B)} \end{align}$

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