Operations among events: union, intersection, difference and complement

First of all, we have to determine what the sample space is. We already know that the possible results are to extract a white ball $(W)$, to extract a red ball $(R)$, to extract a green ball $(G)$ and to extract a black ball $(B)$. So, $\Omega=\lbrace W,R,G,B \rbrace$.

The event $A_1=$"extract a white or red ball" is formed by $A_1= \lbrace W, R \rbrace$. We can see it , considering that $A_1$ is the union of "extract a white ball", $\lbrace W \rbrace$, and "extract a red ball", $\lbrace R \rbrace$.

The event $A_2 =$"extract a ball that is not green" is the opposite of the event "extract a green ball"$= \lbrace G \rbrace$. So, $A_2=\overline{G}$. And so, we know that we can find $A_2$ doing $A_2=\Omega-\lbrace G \rbrace=\lbrace W,R,G,B \rbrace-\lbrace G \rbrace=\lbrace W,R,G \rbrace$.

The event $A_3=$"extract a black ball"$=\lbrace B \rbrace$.

Let's consider now the operations between events that arise next:

$A_1\cup A_3 =$"extract a white or red ball, or to extract a black ball"$= \lbrace W,R \rbrace \cup \lbrace B \rbrace=\lbrace W,R,B \rbrace $.

$A_2-A_1$ is the difference between $A_2$ and $A_1$. It is formed by all the events that are in $A_2$, but not in $A_1$. And so, $A_2-A_1=\lbrace W,R,G \rbrace-\lbrace W,R \rbrace=\lbrace G \rbrace$.

To calculate $\overline{A_1}\cap A_3$, first we have to calculate what $\overline{A_1}$ is. We have seen that the complementary $A_1$ can be found as follows $\overline{A_1}=\Omega - A_1 = \lbrace W,R,G,B \rbrace - \lbrace W,R \rbrace = \lbrace G,B \rbrace.$

Now we can calculate the event $\overline{A_1} \cap A_3$, formed by all the events that satisfy $\overline{A_1}$ and $A_3$. We find it by doing $\overline{A_1}\cap A_3= \lbrace G,B \rbrace \cap \lbrace B \rbrace = \lbrace B \rbrace$.