Answer
The event none of the 4 has some form of mental illness is not unusual.
Work Step by Step
$30$% $=0.3$
The event "18- to 25-year-olds has not any form of mental illness" is the complement of "18- to 25-year-olds has some form of mental illness". So, we use the Complement Rule (see page 275):
$P(18- to~25-year-olds~has~not~any~form~of~mental~illness)=1-P(18- to~25-year-olds~has~some~form~of~mental~illness)=1-0.3=0.7$
The events "18- to 25-year-olds 1 has not any form of mental illness", "18- to 25-year-olds 2 has not any form of mental illness", "18- to 25-year-olds 3 has not any form of mental illness" and "18- to 25-year-olds 4 has not any form of mental illness" are independent.
Now, using the Multiplication Rule (see page 282):
$P(none~of~the~4~has~some~form~of~mental~illness)=P(18- to~25-year-olds~1~has~not~any~form~of~mental~illness)\times P(18- to~25-year-olds~2~has~not~any~form~of~mental~illness)\times P(18- to~25-year-olds~3~has~not~any~form~of~mental~illness)\times P(18- to~25-year-olds~4~has~not~any~form~of~mental~illness)=0.7\times0.7\times0.7\times0.7=0.7^{4}=0.2401$
$0.2401\gt0.05$. The event none of the 4 has some form of mental illness is not unusual.
