Law of total probability - you could have invented this
A knight wants to kill (event ) a dragon. There are two ways to do this: by fighting (event ) the dragon or by outwitting () it. The choice of the way ( or ) is random, and in each case the outcome ( or not ) is also random. For the probability of killing there is a simple, intuitive formula:
This is easy to generalize to the case of many conditioning events. Suppose are mutually exclusive (that is, disjoint) and collectively exhaustive (that is, cover the whole sample space). Then for any event one has
This equation is call the law of total probability.
Application to a sum of continuous and discrete random variables
Let be independent random variables. Suppose that is continuous, with a distribution function , and suppose is discrete, with values . Then for the distribution function of the sum we have
(by independence conditioning on can be omitted)
Compare this to the much more complex derivation in case of two continuous variables.