Question

If you had to construct a mathematical model for events E and F, as described in parts (a) through (e), would you assume that they were independent events? Explain your reasoning.

(a) E is the event that a businesswoman has blue eyes, and F is the event that her secretary has blue eyes.

(b) E is the event that a professor owns a car, and F is the event that he is listed in the telephone book.

(c) E is the event that a man is under 6 feet tall, and F is the event that he weighs more than 200 pounds.

(d) E is the event that a woman lives in the United States, and F is the event that she lives in the Western Hemisphere.

(e) E is the event that it will rain tomorrow, and F is the event that it will rain the day after tomorrow.

Short answer

Answers are

a) E is the event that a businesswoman has blue eyes, and F is the event that her secretary has blue eyes is an independent event

b) E is the event that a professor owns a car, and F is the event that he is listed in the telephone book as an independent event

c) E is the event that a man is under 6 feet tall, and F is the event that he weighs more than 200 pounds is a dependent event.

d) E is the event that a woman lives in the United States, and F is the event that she lives in the Western Hemisphere is a dependent event.

e) E is the event that it will rain tomorrow, and F is the event that it will rain the day after tomorrow is a dependent event.

Step-by-step solution

Given information(part a)

If you had to construct a mathematical model for events E and F, as described in parts (a) through (e), would you assume that they were independent events If E is the event that a businesswoman has blue eyes, and F is the event that her secretary has blue eyes.

Explanation (part a)

Independent

The hiring should not trust the eye stain. We could test the hypothesis that employers are prejudiced to hire someone of the same eye color, but unless that is the objective, we can suppose that a blue eye color of the secretary does not depend on the event that the employer has blue eyes.

Result (part a)

a) E is the event that a businesswoman has blue eyes, and F is the event that her secretary has blue eyes is an independent event.

Given information (part b)

If you had to construct a mathematical model for eventsE andF, as described in parts (a) through (e), would you assume that they were independent events If E is the event that a professor owns a car, and F is the event that he listed in the telephone book.

Explanation(part b)

Independent

These two events do not have a direct connection. The independence can be questionable because telephone book lists more older people, and a lesser percentage of older people drive.

Result (part b)

a) E is the event that a businesswoman has blue eyes, and F is the event that her secretary has blue eyes is an independent event.

Given information (part c)

If you had to construct a mathematical model for events E and F, as described in parts (a) through (e), would you assume that they were independent events if E is the event that a man is under 6 feet tall, and F is the event that he weighs more than 200 pounds.

Explanation (part c)

Dependent

A lesser percentage of men under 6 ft tall weights over 200 pounds. Mathematically:

P( weights over 200 under 6ft tall )< P( weights over 200)

$\Rightarrow$P(weights over 200, and is under 6ft tall )< P (weights over 200) P( weights over 200)

Result (Part c)

(c) E is the event that a man is under 6 feet tall, and F is the event that he weighs more than 200 pounds is a dependent event.

Given information (part d)

If you had to construct a mathematical model for events E and F, as described in parts (a) through (e), would you assume that they were independent events if E is the event that a woman lives in the United States, and F is the event that she lives in the Western Hemisphere.

Explanation (part d)

Dependent

The United States is in the Western hemisphere :

P( she lives in the US and doesn't live in the Western Hemisphere )=0

P( she lives in the US ) P( doesn't live in the Western Hemisphere )>0

These two are not equal, therefore events that a woman lives in the US and doesn't live in the Western Hemisphere are dependent so the complement of the latter - that a woman lives in the Western Hemisphere and the former - that she lives in the US are dependent.

Result (Part d)

(d) E is the event that a woman lives in the United States, and F is the event that she lives in the Western Hemisphere is a dependent event.

Given information (part e)

If you had to construct a mathematical model for events E and F, as described in parts (a) through (e), would you assume that they were independent events if E is the event that it will rain tomorrow, and F is the event that it will rain the day after tomorrow.

Explanation (Part e)

Dependent

Rains drop's because excessively largely water collects in the clouds. So if some of the water drains from the stratosphere (it showers), there shouldn't be sufficient water left for another showery day.

P(rains the day after tomorrow | rains tomorrow )<P(rains the day after tomorrow )

$\Rightarrow$P( rains tomorrow and the day after )<P( rains the day after tomorrow ) P(rains tomorrow )

Result (part e)

(e) E is the event that it will rain tomorrow, and F is the event that it will rain the day after tomorrow is a dependent event