Question

An alarming number of U.S. adults are either overweight or obese. The distinction between overweight and obese is made on the basis of body mass index (BMI), expressed as weight/height 2. An adult is considered overweight if the BMI is 25 or more but less than 30. An obese adult will have a BMI of 30 or greater. According to a January 2012 article in the Journal of the American Medical Association, 33.1% of the adult population in the United States is overweight and 35.7% is obese.

A. What is the probability that a randomly selected adult is either overweight or obese?
B. What is the probability that a randomly selected adult is neither overweight or obese?
C. Are the events "overweight and "obese" exhaustive?

Answer 1

A- 0.688.

B- 0.312.

C- yes

Answer 2

(A) The probability that a randomly selected adult is either overweight or obese is 0.688.

(B) The probability that a randomly selected adult is neither overweight nor obese is 0.312.

(C) Yes, the events "overweight" and "obese" are exhaustive.

Explanation:

We are given that an adult is considered overweight if the BMI is 25 or more but less than 30. An obese adult will have a BMI of 30 or greater.

According to a January 2012 article in the Journal of the American Medical Association, 33.1% of the adult population in the United States is overweight and 35.7% is obese.

Let the event , A = a person is overweight

B = a person is obese.

So according to the question;

The probability that an adult population in the United States is overweight = P(A) = 0.331

The probability that an adult population in the United States is obese = P(B) = 0.357

(A) As we know that the events of a person being overweight or obese cannot occur together.

This means P(A ∩ B) = 0.

Now, the probability that a randomly selected adult is either overweight or obese is given by = P(A
`\bigcup` B)

P(A
`\bigcup` B) = P(A) + P(B) - P(A ∩ B)

= 0.331 + 0.357 - 0

= 0.688

Hence, the probability that a randomly selected adult is either overweight or obese is 0.688.

(B) Now, the probability that a randomly selected adult is neither overweight nor obese is given by = P(A'
`\bigcup` B')

P(A'
`\bigcup` B') = 1 - P(A
`\bigcup` B)

= 1 - 0.688 = 0.312

Hence, the probability that a randomly selected adult is neither overweight nor obese is 0.312.

(C) Exhaustive events are those events which cannot occur together, but they form a sample space when combined together.

In our question also, the person can't be both overweight and obese at the same time, and together the event of a person being overweight or obese forms a sample space of people who are heavier.

Hence, the events "overweight" and "obese" are exhaustive.