The probability of selecting an appointment with patients under 18 years old is
[tex]P=\frac{6+9}{6+4+9+1}=\frac{15}{20}=\frac{3}{4}[/tex]Because there are 15 patients under 18 years old out of 20 patients in total.
The probability of regular cleaning appointments is
[tex]P=\frac{6+4}{6+4+9+1}=\frac{10}{20}=\frac{1}{2}[/tex]Because there are 10 regular cleaning appointments out of 20.
Then, we multiply both probabilities to find the answer
[tex]P=\frac{3}{4}\cdot\frac{1}{2}=\frac{3}{8}[/tex]