The conditional probability of
given that
has occurred, denoted
, equals
![\begin{displaymath}
P(A\vert B) = {P(A\cap B)\over P(B)},
\end{displaymath}](c2_953.gif) |
(1) |
which can be proven directly using a Venn Diagram. Multiplying through, this becomes
![\begin{displaymath}
P(A\vert B)P(B) = P(A\cap B),
\end{displaymath}](c2_954.gif) |
(2) |
which can be generalized to
![\begin{displaymath}
P(A\cup B\cup C) =P(A)P(B\vert A)P(C\vert A\cup B).
\end{displaymath}](c2_955.gif) |
(3) |
Rearranging (1) gives
![\begin{displaymath}
P(B\vert A) = {P(B\cap A)\over P(A)}.
\end{displaymath}](c2_956.gif) |
(4) |
Solving (4) for
and plugging in to (1) gives
![\begin{displaymath}
P(A\vert B) = {P(A)P(B\vert A)\over P(B)}.
\end{displaymath}](c2_958.gif) |
(5) |
See also Bayes' Formula
© 1996-9 Eric W. Weisstein
1999-05-26