In this section we learn how to calculate the probability of \(A\) and \(B\) occuring, that's when both events \(A\) and \(B\) happen at the same time and we learn the formula for independent events \(A\) and \(B\). Although we sometimes refer to this probability with the notation \(p\begin{pmatrix}A \ \text{and} \ B \end{pmatrix}\) we'll use the correct mathematical notation and write: \[p\begin{pmatrix}A \cap B \end{pmatrix}\] To be clear, \(p\begin{pmatrix}A \cap B \end{pmatrix}\) should be read "the probability of A and B occurring".
The Venn diagram shown here shows two sets \(A\) and \(B\), which correspond to two events \(A\) and \(B\). The two sets overlap, we say they intersect, thereby creating the set \(A\cap B\) (that's the event "\(A\) and \(B\)").
If we're given, or can find, how many elements are inside the set \(A \cap B\) and we know how many elements are inside the universal set \(U\), that's \(n\begin{pmatrix}U \end{pmatrix}\), then we can calculate \[p\begin{pmatrix}A \cap B \end{pmatrix} = \frac{n \begin{pmatrix} A \cap B \end{pmatrix}}{n \begin{pmatrix}U\end{pmatrix}}\] We frequently won't be given enough information to find both \(n\begin{pmatrix}A \cap B \end{pmatrix}\) and \(n\begin{pmatrix} U \end{pmatrix}\). Instead we'll "just" be given, or will be able to find the probabilities \(p\begin{pmatrix}A\end{pmatrix}\) and \(p\begin{pmatrix}B \end{pmatrix}\). In such cases we'll have to use the formula that we learn next.
The formula we learn here is for independent events. If you're not sure what independent events are then do make sure to click on the "Learn More" button below.
Given two events \(A\) and \(B\) that are independent, then provided we are given (or can find) the probabilities of both events, \(p\begin{pmatrix}A \end{pmatrix}\) and \(p\begin{pmatrix}B \end{pmatrix}\), we can calculate the probability \(p\begin{pmatrix}A\cap B \end{pmatrix}\) using the following formula
Given two independent events \(A\) and \(B\), the probability of the event "\(A\) and \(B\)" occurring is written \(p\begin{pmatrix}A \cap B \end{pmatrix}\) and is given by the formula: \[p\begin{pmatrix}A \cap B \end{pmatrix} = p\begin{pmatrix}A \end{pmatrix} \times p\begin{pmatrix}B \end{pmatrix}\]
The following tutorial we see, through a couple of examples, how to use the multplcation rule for independent events.
For a survey, a group of people, with pets at home, was asked whether they had a cat or a dog. The result is summarized in the Venn diagram, shown here.
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