Question

​In a sample of 800 U.S.​ adults, 199 think that most celebrities are good role models. Two Two U.S. adults are selected at random from the population of all U.S. adults without replacement. Assuming the sample is representative of all U.S.​adults, 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 ________?(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 _______?(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 ______? (Round to three decimal places as​ needed.)

Expert Answer

Answer 1

Answer:

Step-by-step explanation:

Given that in a sample of 800 U.S.​ adults, 199 think that most celebrities are good role models. Two Two U.S. adults are selected at random from the population of all U.S. adults without replacement.

In 800 adults 199 are favourable and remaining 601 are against.

a) The probability that both adults think most celebrities are good role models is ________= Prob of selecting both from 199

= $\frac{199C2}{800C2} \\=0.0616$

=0.062

b) The probability that neither adult thinks most celebrities are good role models is _______=P(both selecting from 601)

=  $\frac{601C2}{800C2} \\=0.5641$

=0.564

c) The probability that at least one of the two adults thinks most celebrities are good role models ______

=1-Prob that neither thinks

= 1- 0.564

= 0.436

Answer 2

Using the hypergeometric distribution, it is found that:

a) The probability that both adults think most celebrities are good role models is 0.062.

b) The probability that neither adult thinks most celebrities are good role models is 0.564.

c) The probability that at least one of the two adults thinks most celebrities are good role models 0.436.

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The adults are chosen without replacement, thus, the hypergeometric distribution is used to solve this question.

Hypergeometric distribution:

The probability of x successes is given by the following formula:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}$

In which:

• x is the number of successes.
• N is the size of the population.
• n is the size of the sample.
• k is the total number of desired outcomes.

Combinations formula:

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

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• 800 adults means that $N = 800$
• 199 think that most celebrities are role models, thus $k = 199$
• Sample of 2, thus $n = 2$

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Item a:

This probability is P(X = 2), thus:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}$

$P(X = 2) = h(2,800,2,199) = \frac{C_{199,2}*C_{601,0}}{C_{800,2}} = 0.062$

The probability that both adults think most celebrities are good role models is 0.062.

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Item b:

This probability is P(X = 0), thus:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}$

$P(X = 0) = h(0,800,2,199) = \frac{C_{199,0}*C_{601,2}}{C_{800,2}} = 0.564$

The probability that neither adult thinks most celebrities are good role models is 0.564.

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Item c:

This is:

$P(X \geq 1) = 1 - P(X = 0) = 1 - 0.564 = 0.436$

The probability that at least one of the two adults thinks most celebrities are good role models 0.436.

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