L06.5 Total Expectation Theorem
An important reason why conditional probabilities are very useful is that they allow us to divide and conquer. They allow us to split complicated probability modes into simpler submodels that we can then analyze one at a time. Let me remind you of the Total Probability Theorem that has his particular flavor. We divide our sample space into three disjoint events-- A1, A2, and A3. And these events form a partition of the sample space, that is, they exhaust all possibilities. They correspond to three alternative scenarios, one of which is going to occur. And then we may be interested in a certain event B. That event B may occur under either scenario. And the Total Probability Theorem tells us that we can calculate the probability of event B by considering the probability that it occurs under any given scenario and weigh those probabilities according to the probabilities of the different scenarios. Now, let us bring random variables into the picture. Let us fix a particular v...