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# Total probability theorem

Probability and Statistics
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Probability Theory
schedule Jul 1, 2022
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The total probability theorem is used to calculate the probability of an outcome, which may occur under several different scenarios.

It states that:

$$\mathbb{P}(B) = \sum_i \mathbb{P}(A_i)\, \mathbb{P}(B|A_i)$$

In English, it states that the probability of an event $\mathbb{P}(B)$ is the sum of its weighted conditional probabilities.

Remember from the multiplication rule that the probability of two events $A$ and $B$ both occurring is$\mathbb{P}(A\cap B)=\mathbb{P}(A)\cdot \mathbb{P}(B|A)$. With the total probability theorem, all we are stating is that if there are several scenarios $A_i$, then to calculate the probability of $B$ we just have to apply the multiplication rule for each scenario $A_i$ and sum them all together.

# Example

We want to calculate the probability of a person going out given the following probabilities and scenarios:

From the total probability theorem we know that:

\begin{align*} \mathbb{P}(\text{Go out}) &= \mathbb{P}(\text{Sunny})\,\mathbb{P}(\text{Go out}|\text{Sunny}) + \mathbb{P}(\text{Rainy})\,\mathbb{P}(\text{Go out}|\text{Rainy}) + \mathbb{P}(\text{Cloudy})\,\mathbb{P}(\text{Go out}|\text{Cloudy})\\ &= 0.3 * 0.7 + 0.2 * 0.2 + 0.5*0.5 \\ &= 0.5 \end{align*}
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