Question

in a sample of 1200 u.s. adults, 186 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 0.024. (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 0.714. (round to three decimal places as needed.)

Explanation:

Step1: Calculate probability of one adult thinking most celebrities are good role - models

The proportion of adults who think most celebrities are good role - models in the sample is $p=\frac{186}{1200}=0.155$.

Step2: Calculate probability for part (a)

The probability that both adults think most celebrities are good role - models (using the multiplication rule for independent events, but since it's without replacement, we approximate for large sample. The probability of the first adult having the view and then the second) is $P(\text{both})=\frac{186}{1200}\times\frac{185}{1199}\approx0.024$.

Step3: Calculate probability for part (b)

The probability that neither adult thinks most celebrities are good role - models. The probability that one adult doesn't think so is $1 - 0.155 = 0.845$. So the probability that neither does is $P(\text{neither})=0.845\times0.844\approx0.714$.

Step4: Calculate probability for part (c)

The probability that at least one adult thinks most celebrities are good role - models is $P(\text{at least one}) = 1 - P(\text{neither})=1 - 0.714 = 0.286$.

Answer:

(a) $0.024$
(b) $0.714$
(c) $0.286$