Respuesta :
Answer:
(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 [tex]\bigcup[/tex] B)
P(A [tex]\bigcup[/tex] 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' [tex]\bigcup[/tex] B')
P(A' [tex]\bigcup[/tex] B') = 1 - P(A [tex]\bigcup[/tex] 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.
