DES - Theorem 'Total Probability Rule'

Summary

The probability that we are at state $A$ is equal to the sum of all probabilities that the events $B_i$ brings us to state $A$ multiplied by the probability that event $B_i$ has to happen.

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Total Probability Rule

Let $\Omega$ be the set of all possible outcomes of a random experiment (also called the sample space) Let $A$ be an event (i.e., a subset of $\Omega$ ), and $B_1,\dots ,B_n$ be a partition of $\Omega$ :

• $B_1,\dots ,B_n$ are events
• $B_1\cup \dots \cup B_n=\Omega$:
• $B_i\cap B_j=\varnothing$; for all $i\ne j$

Then:

$$P(A)=\sum _{i=1}^{n}P\left(A\mid B_i\right)\cdot P\left(B_i\right)$$

If $C$ is another event, we also have:

$$P(A\mid C)=\sum _{i=1}^{n}P\left(A\mid B_i,C\right)\cdot P\left(B_i\mid C\right)$$

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Observation:

Usually we use this theorem because computing the probability of the sum is much easier than directly computing $P(A)$ or $P(A\mid C)$ directly. Practically we get to use the additional information provided by $B_i$

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~Example: