Answer: 0.420
Explanation:
Given: The proportion of 18- to 25-year-olds have some form of mental illness: p= 0.30
Let x be the number of 18- to 25-year-olds that have mental illness.
A person can either have illness or not , so it follows Binomial distribution.
Probability distribution function:
[tex]P(X=x) =\ ^nC_xp^x(1-p)^x[/tex] , where n= samples oize , p probability of success in each trial , x= number of successes.
Here, n=6 ,
Required probability:
[tex]P(x\leq 1)=P(x=0)+P(x=1)\\\\=\ ^6C_0(0.30)^0(0.70)^6+^6C_1(0.30)^1(0.70)^5\\\\=(1)(1)(0.70)^6+(6)(0.30)^1(0.70)^5\\\\=0.420175\approx0.420[/tex]
Hence, the probability at least one of six randomly selected 18- to 25-year-olds have some form of mental illness = 0.420
