The community development opens sport ground for youth to participate in sport based on soccer and netball. 20 students play soccer, 18 play netball, 8 play both, and 5 play neither. Draw a Venn diagram, find the total number of students, and calculate probabilities related to student participation.

22.2.1 Venn diagram regions

Let $S$ be the set of students who play soccer and $N$ be the set of students who play netball.

Given:
$|S| = 8$
$|N| = 20$
$|S \cap N| = 8$ (both)
Neither: $5$

Step 1: Only soccer
$|S \setminus N| = |S| - |S \cap N|$
Substitute values:
$|S \setminus N| = 8 - 8 = 0$

Step 2: Only netball
$|N \setminus S| = |N| - |S \cap N|$
Substitute values:
$|N \setminus S| = 20 - 8 = 12$

Step 3: Both sports
$|S \cap N| = 8$ (given)

Step 4: Soccer or netball (union)
$|S \cup N| = |S| + |N| - |S \cap N|$
Substitute values:
$|S \cup N| = 8 + 20 - 8 = 20$

Venn diagram:

• Soccer only: $0$
• Netball only: $12$
• Both: $8$
• Outside (neither): $5$

22.2.2 Total students in the school
Step 5: Total students
Total = $|S \cup N|$ + neither
$25 = 20 + 5$
25 students
25

22.2.3 Students who play soccer or netball, but not both
Only soccer + only netball
$0 + 12 = 12$
12 students
12

22.2.4 Probability a randomly selected student plays soccer or netball but not both
Number who play soccer or netball but not both = $12$
Total students = $25$
$P = \frac{12}{25}$
\frac{12}{25}
\dfrac{12{25}}

22.2.5 Probability a randomly selected student plays neither sport
Number who play neither = $5$
Total students = $25$
$P = \frac{5}{25}$
Simplify:
Divide numerator and denominator by $5$
$P = \frac{1}{5}$
\frac{1}{5}
\dfrac{1{5}}