Answer
$P(at~least~one~of~six~18- to~25-year-olds~has~some~form~of~mental~illness)=0.882351$
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", "18- to 25-year-olds 4 has not any form of mental illness", "18- to 25-year-olds 5 has not any form of mental illness" and "18- to 25-year-olds 6 has not any form of mental illness" are independent.
Now, using the Multiplication Rule (see page 282):
$P(none~of~the~6~has~some~form~of~mental~illness)=P(18- to~25-year-olds~1~has~not~any~form~of~mental~illness)\times P(18- to~25-year-olds~2~has~not~any~form~of~mental~illness)\times P(18- to~25-year-olds~3~has~not~any~form~of~mental~illness)\times P(18- to~25-year-olds~4~has~not~any~form~of~mental~illness)\times P(18- to~25-year-olds~5~has~not~any~form~of~mental~illness)\times P(18- to~25-year-olds~6~has~not~any~form~of~mental~illness)=0.7\times0.7\times0.7\times0.7\times0.7\times0.7=0.7^{6}=0.117649$
The event "at least one of six 18- to 25-year-olds has some form of mental illness" is the complement of "none of the 6 has some form of mental illness". So, we use the Complement Rule (see page 275):
$P(at~least~one~of~six~18- to~25-year-olds~has~some~form~of~mental~illness)=1-P(none~of~the~6~has~some~form~of~mental~illness)=1-0.117649=0.882351$
