### Chapter 3. Probability: Probability

### Law of Total Probability

Partition

A **partition **of the sample space #\Omega# for an experiment is any collection of mutually exclusive events #A_1, A_2, \ldots, A_k# whose union #A_1 \cup A_2 \cup \ldots \cup A_k# equals #\Omega#.

This means that every possible outcome of the experiment is an element of one and only one of the events.

Law of Total Probability

Given a partition #A_1, A_2, \ldots, A_k# and some other event #B#, it is possible to write #B# as:

\[B = (A_1 \cap B) \cup (A_2 \cap B) \cup \ldots \cup (A_k \cap B)\]

Then

\[\begin{array}{rcl}

\mathbb{P}(B) &=& \mathbb{P}[(A_1 \cap B) \cup (A_2 \cap B) \cup \ldots \cup (A_k \cap B)]\\

&=& \mathbb{P}(A_1 \cap B) + \mathbb{P}(A_2 \cap B) + \ldots + \mathbb{P}(A_k \cap B)\\

&=& \mathbb{P}(B\,|\,A_1)\cdot \mathbb{P}(A_1) + \mathbb{P}(B\,|\,A_2)\cdot \mathbb{P}(A_2) + \ldots + \mathbb{P}(B\,|\,A_k)\cdot \mathbb{P}(A_k)

\end{array}\]

provided #\mathbb{P}(A_i) \neq 0# for each #i#.

This is called the **Law of Total Probability**.

Your parents will travel on vacation in the summer to either Spain, Italy, France, or Croatia with respective probabilities #0.2, 0.3, 0.1, \text{and } 0.4#.

Depending on which country they choose, you will join them with respective probabilities #0.4, 0.6, 0.8, \text{and } 0.3#.

What is the probability that you will join your parents on vacation this summer? Round your answer to #2# decimal places.

Define the following events:

- #A_1 =# Your parents choose Spain
- #A_2 =# Your parents choose Italy
- #A_3 =# Your parents choose France
- #A_4 =# Your parents choose Croatia
- #B\,\,=\,# You join your parents

\[\begin{array}{rcl}

\mathbb{P}(B) &=& \mathbb{P}(B\,|\,A_1)\cdot \mathbb{P}(A_1) + \mathbb{P}(B\,|\,A_2)\cdot \mathbb{P}(A_2) + \mathbb{P}(B\,|\,A_3)\cdot \mathbb{P}(A_3) + \mathbb{P}(B\,|\,A_4)\cdot \mathbb{P}(A_4)\\

&=& 0.4 \cdot 0.2 + 0.6 \cdot 0.3 + 0.8 \cdot 0.1 + 0.3 \cdot 0.4\\

&=& 0.46\\

\end{array}\]

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