## 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:

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Its derivation is straightforward from the definition of conditional probability: since and cover the whole sample space and are disjoint, we have by additivity of probability

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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

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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)

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Compare this to the much more complex derivation in case of two continuous variables.

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