(Combined Events Part 1)

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:\To be clear, \(p\begin{pmatrix}A \cap B \end{pmatrix}\) should be read "the probability of A and B occurring".

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Venn Diagram for \(p\begin{pmatrix}A \cap B \end{pmatrix}\)

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 \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.


Formula for \(p\begin{pmatrix} A \cap B\end{pmatrix}\) - Independent Events

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.


Independent Events, or Dependent Events?

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When considering two events or more, it is important to establish whether we"re dealing with independent events or dependent events. This is important as it will affect the formula we use and therefore the results we obtain.

Independent Events

Two events \(A\) and \(B\) are independent if the occurrence of either one of the two events has absoluetly no impact on the likelihood of the other event occurring.

For instance the two events \(A\) and \(B\) defined as:\(A\): Charlotte does well at her Math Test\(B\): it rains in Stockholmare independent. Whether or not it rains in Stockholm will have no impact on how well Charlotte does at her Math Test. Similarly, whether Charlotte does wll at her Math Test or not has no impact on whether or not it will rain in Stockholm.

Dependent Events (Conditional Probability)

Two events \(A\) and \(B\) are dependent if the occurrence of one of them impacts the likelihood of the other occurring.

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For example, the two events \(A\) and \(B\) defined as:\(A\): Clara leaves home with an umbrella.\(B\): The weather forecast predicts rainfall.are dependent. Indeed, whether or not it rains is more than likely to have an impact on whether or not Clara takes an umbrella with her.