Question

According to the Department of Health and Human

Services, 30% of 18- to 25-year-olds have some form
of mental illness.
What is the probability AT LEAST one of six randomly
selected 18- to 25-year-olds have some form of mental
illness? Round to the nearest thousandth.

Answer 1

Answer: 0.420

Step-by-step 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:

`P(X=x) =\ ^nC_xp^x(1-p)^x` , where n= samples oize , p probability of success in each trial , x= number of successes.

Here, n=6 ,

Required probability:

`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`

Hence, the probability at least one of six randomly selected 18- to 25-year-olds have some form of mental illness = 0.420