Let x represent those who both football and basketball
The given information can be illustrated in a Venn diagram as shown in the attachment.
We solve the equation below to find the value of x.
[tex](53-x)+x+(24-x)+31=101[/tex]
[tex]\implies -x+x-x=101-53-31-24[/tex]
[tex]\implies -x=-7[/tex]
[tex]\implies x=7[/tex]
From the second diagram;
25. [tex]P(Basketball)=\frac{17}{101}[/tex]
26. [tex]P(Football)=\frac{46}{101}[/tex]
27. [tex]P(Football\: \cap\:Basketball)=\frac{7}{101}[/tex]. This is because 7 play both Football and Basketball.
28. [tex]P(Football\: \cup\:Basketball)=\frac{46}{101}+\frac{17}{101}-\frac{7}{101}=\frac{56}{101}[/tex]. This is because there is intersection.
29. [tex]P(Neither\: \cup\:Both)=\frac{31}{101}+\frac{7}{101}=\frac{38}{101}[/tex]. The two events are mutually exclusive.
