Questions: In a sample of 800 U.S. adults, 191 think that most celebrities are good role models. Two U.S. adults are selected from this sample without replacement. Complete parts (a) through (c). (a) Find the probability that both adults think most celebrities are good role models. The probability that both adults think most celebrities are good role models is .057 (Round to three decimal places as needed.) (b) Find the probability that neither adult thinks most celebrities are good role models. The probability that neither adult thinks most celebrities are good role models is .579 (Round to three decimal places as needed.) (c) Find the probability that at least one of the two adults thinks most celebrities are good role models. The probability that at least one of the two adults thinks most celebrities are good role models is (Round to three decimal places as needed.)

Solution Steps

To find the probability that both selected adults think most celebrities are good role models, we use the hypergeometric distribution formula:

\[ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \]

where:

• \(N = 800\) (total number of adults),
• \(K = 191\) (number of adults who think most celebrities are good role models),
• \(n = 2\) (number of adults selected),
• \(k = 2\) (number of adults in the sample who think most celebrities are good role models).

Calculating this gives:

\[ P(X = 2) = \frac{\binom{191}{2} \binom{609}{0}}{\binom{800}{2}} = 0.057 \]

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

\[ \boxed{0.057} \]

Next, we calculate the probability that neither of the selected adults thinks most celebrities are good role models. This is also done using the hypergeometric distribution:

\[ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \]

where:

• \(N = 800\),
• \(K = 609\) (number of adults who do not think most celebrities are good role models),
• \(n = 2\),
• \(k = 0\) (number of adults in the sample who think most celebrities are good role models).

Calculating this gives:

\[ P(X = 0) = \frac{\binom{609}{2} \binom{191}{0}}{\binom{800}{2}} = 0.579 \]

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

\[ \boxed{0.579} \]

To find the probability that at least one of the two selected adults thinks most celebrities are good role models, we can use the complement rule:

\[ P(\text{at least one}) = 1 - P(\text{none}) \]

Substituting the previously calculated probability:

\[ P(\text{at least one}) = 1 - 0.579 = 0.421 \]

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

\[ \boxed{0.421} \]

Final Answer