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 #0.1, 0.4, 0.3, \text{and } 0.2#.
Depending on which country they choose, you will join them with respective probabilities #0.8, 0.4, 0.3, \text{and } 0.6#.
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
\[\begin{array}{rcl}
\mathbb{P}(B) &=& \mathbb{P}(B\,|\,A_1)\cdot \mathbb{P}(A_1) + \mathbb{P}(B\,|\,A_2)\cdot \mathbb{P}(A_2) + \mathbb{P}(B\,|\,A_3)\cdot \mathbb{P}(A_3) + \mathbb{P}(B\,|\,A_4)\cdot \mathbb{P}(A_4)\\
&=& 0.8 \cdot 0.1 + 0.4 \cdot 0.4 + 0.3 \cdot 0.3 + 0.6 \cdot 0.2\\
&=& 0.45\\
\end{array}\]
