Answer
$P(all~the~6~have~some~form~of~mental~illness)=0.000729$
Work Step by Step
$30$% $=0.3$
The events "18- to 25-year-olds 1 has some form of mental illness", "18- to 25-year-olds 2 has some form of mental illness", "18- to 25-year-olds 3 has some form of mental illness", "18- to 25-year-olds 4 has some form of mental illness", "18- to 25-year-olds 5 has some form of mental illness" and "18- to 25-year-olds 6 has some form of mental illness" are independent.
Now, using the Multiplication Rule (see page 282):
$P(all~the~6~have~some~form~of~mental~illness)=P(18- to~25-year-olds~1~has~some~form~of~mental~illness)\times P(18- to~25-year-olds~2~has~some~form~of~mental~illness)\times P(18- to~25-year-olds~3~has~some~form~of~mental~illness)\times P(18- to~25-year-olds~4~has~some~form~of~mental~illness)\times P(18- to~25-year-olds~5~has~some~form~of~mental~illness)\times P(18- to~25-year-olds~6~has~some~form~of~mental~illness)=0.3\times0.3\times0.3\times0.3\times0.3\times0.3=0.3^{6}=0.000729$
