Law of Total Probability

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 $0.3, 0.4, 0.2, \text{and } 0.1$ .

Depending on which country they choose, you will join them with respective probabilities $0.5, 0.2, 0.7, \text{and } 0.4$ .

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

$\mathbb{P}(\text{You join your parents})= 0.41$

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

Then
$\begin{array}{rcl} \mathbb{P}(B) &=& \mathbb{P}(B\,|\,A_1)\cdot \mathbb{P}(A_1) + \mathbb{P}(B\,|\,A_2)\cdot \mathbb{P}(A_2) + \mathbb{P}(B\,|\,A_3)\cdot \mathbb{P}(A_3) + \mathbb{P}(B\,|\,A_4)\cdot \mathbb{P}(A_4)\\ &=& 0.5 \cdot 0.3 + 0.2 \cdot 0.4 + 0.7 \cdot 0.2 + 0.4 \cdot 0.1\\ &=& 0.41\\ \end{array}$

New example