Law of Total Probability

Total Probability Theorem

Let's partition our sample space \(\Omega\) into \(A_1, A_2\) and \(A_3\), as shown in the figure below.

Then, for any set \(B\), we have

\[\begin{align} \mathbf{P}(B)&=\mathbf{P}(A_1\cap B)+\mathbf{P}(A_2\cap B)+\mathbf{P}(A_3\cap B)\\ &=\mathbf{P}(A_1)\,\mathbf{P}(B\lvert A_1)+\mathbf{P}(A_2)\,\mathbf{P}(B\lvert A_2)+\mathbf{P}(A_3)\,\mathbf{P}(B\lvert A_3). \end{align}\]

In general,

\[\mathbf{P}(B)=\sum_{i}\mathbf{P}(A_i)\,\mathbf{P}(B\lvert A_i).\]