Question

Your neighborhood homeowners’ association (HOA) conducted a survey to collect data about people in your neighborhood. Based on the survey results, the HOA will decide what to spend money on. Here are the results of the survey:

• Between one-fourth and one-third of the people in your neighborhood are younger than eighteen years of age.

• Between one-half and two-thirds of the people are over the age of forty.

• Between one-sixth and one-fifth of the people in your neighborhood golf on Saturdays in the summer. Between two-thirds and three-fourths of the people in your neighborhood go to the pool on Saturdays in the summer. The rest prefer indoor activities.

• Your neighborhood has 120 people.

What are the greatest and least numbers of people who are between the ages of eighteen and forty and who prefer indoor activities?

Answer 1

1/4 < P(Age < 18) < 1/3

1/2 < P(Age > 40) < 2/3

P(Age between 18 and 40) = 1 - P(Age < 18) - P(Age > 40)

Thurs,

1 - 1/3 - 2/3 < P(Age between 18 and 40) < 1 - 1/4 - 1/2

0 < P(Age between 18 and 40) < 1/4

Similarly,

1/6 < P(golf) < 1/5

2/3 < P(pool) < 3/4

P(indoor) = 1 - P(golf) - P(pool)

1 - 1/5 - 3/4 < P(indoor) < 1 - 1/6 - 2/3

1/20 < P(indoor) < 1/6

The greatest probability of people who are between the ages of eighteen and forty and who prefer indoor activities

= P(Age between 18 and 40 and indoor) = maximum of maximum value of P(Age between 18 and 40) and P(indoor)

= 1/4

The least probability of people who are between the ages of eighteen and forty and who prefer indoor activities

= P(Age between 18 and 40 and indoor) = 0 (if the events ages of eighteen and forty and prefer indoor activities are disjoint events)

Thus,greatest and least numbers of people who are between the ages of eighteen and forty and who prefer indoor activities are 120 * 1/4 = 30 and 0 respectively.