In a sample of 1000 U.S. adults. 217 think that most celebrities are good role models. Two U.S. adults are selected from this sample without replacement.


Find the probability that both adults think most celebrities are good role models

(Round to three decimal places as needed.)


Find the probability that neither adult thinks most celebrities are good role models

(Round to three decimal places as needed.)


Find the probability that at least one of the two adults thinks most celebrities are good role models

(Round to three decimal places as needed)

Answer:

(a) 0.047

(b) 0.613

(c) 0.387

Step-by-step explanation:

In a sample of 1000 U.S. adults 217 think that most celebrities are good role models.

The proportion of U.S. adults who think that most celebrities are good role models is, p = 0.217.

Two U.S. adults are selected from this sample without replacement.

Let X denote the number of U.S. adults who think that most celebrities are good role models.

Both the individuals are independent of each other.

The random variable X follows a binomial distribution with parameters n = 2 and p = 0.217.

The probability mass function of X is:

[tex]P(X=x)={2\choose x}(0.217)^{x}(1-0.217)^{2-x};x=0,1,2[/tex]

(a)

Compute the probability that both adults think most celebrities are good role models as follows:

[tex]P(X=2)={2\choose 2}(0.217)^{2}(1-0.217)^{2-2}\\=1\times 0.047089\times 1\\=0.047[/tex]

Thus, the probability that both adults think most celebrities are good role models is 0.047.

(b)

Compute the probability that neither adult thinks most celebrities are good role models as follows:

[tex]P(X=0)={2\choose 0}(0.217)^{0}(1-0.217)^{2-0}\\=1\times 1\times 0.613089\\=0.613[/tex]

Thus, the probability that neither adult thinks most celebrities are good role models is 0.613.

(c)

Compute the probability that at least one of the two adults thinks most celebrities are good role models  as follows:

[tex]P(X\geq 1)=1-P(X<1)\\=1-P(X=0)\\=1-0.613\\=0.387[/tex]

Thus, the probability that at least one of the two adults thinks most celebrities are good role models  is 0.387

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