Question

In a sample of 1100 U.S.​ adults, 208 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.
​(b) Find the probability that neither adult thinks most celebrities are good role models.
​(c) Find the probability that at least one of the two adults thinks most celebrities are good role models.

Answer 1

a) 3.56% probability that both adults think most celebrities are good role models.

b) 65.74% probability that neither adult thinks most celebrities are good role models.

c) 34.26% probability that at least one of the two adults thinks most celebrities are good role models.

Explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

The order in which the adults are chosen is not important. So we use the combinations formula to solve this question.

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) = (n!)/(x!(n-x)!)`

​(a) Find the probability that both adults think most celebrities are good role models.

Desired outcomes:

Think most celebrities are good role models, so 2 from a set of 208. Then

`D = C_(208,2) = (208!)/(2!(208-2)!) = 21528`

Total outcomes:

Two adults from a set of 1100. Then

`T = C_(1100,2) = (1100!)/(2!(1100-2)!) = 604450`

Probability:

`p = (D)/(T) = (21528)/(604450) = 0.0356`

3.56% probability that both adults think most celebrities are good role models.

(b) Find the probability that neither adult thinks most celebrities are good role models.

Desired outcomes:

2 from a set of 1100 - 208 = 892. Then

`D = C_(892,2) = (892!)/(2!(892-2)!) = 397386`

Total outcomes:

Two adults from a set of 1100. Then

`T = C_(1100,2) = (1100!)/(2!(1100-2)!) = 604450`

Probability:

`p = (D)/(T) = (397386)/(604450) = 0.6574`

65.74% probability that neither adult thinks most celebrities are good role models.

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

Either neither think, or at least one does. The sum of the probabilities of these events is 100%.

From b), 65.74% probability that neither think.

So

65.74 + p = 100

p = 100 - 65.74

p = 34.26

34.26% probability that at least one of the two adults thinks most celebrities are good role models.