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?

(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.

Step-by-step 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 B)

         P(A 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' B')

              P(A' B') = 1 - P(A 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.

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step-by-step explanation: