We now learn about the total probability formula, for conditional probability.
Given two events \(A\) and \(B\), such that the probability of \(A\) is affected by whether or not event \(B\) has occurred, then to calculate the probability of event \(A\) occuring we need to consider the following two possible mutually exclusive events:
Given two events \(A\) and \(B\), if the probability of event \(A\) is affected by whether or not event \(B\) occors then we can calculate the probability of event \(A\) occuring using:
\[p(A) = p(B)\times p\begin{pmatrix} A | B \end{pmatrix} + p(B')\times p\begin{pmatrix} A | B' \end{pmatrix} \]
This formula is "saying":
The total probability formula makes a litle more sense when we read it off a tree diagram.
We do this here, highlighting the key paths:
Looking at this tree diagram, we observe starting from the left:
In the following tutorial we learn how the total probability formula can be derived using a tree diagram as well as watch a worked example. Watch it now.