Answer :
To tackle this problem, let's proceed with each part step-by-step:
(a) First, we need to find the probability of a randomly selected American being an overweight man. Since we know the percentage of men who are overweight and the percentage of the population that are men, we can calculate this by multiplying the two probabilities together:
The probability that an individual is a man (P_man) is 48.8%, or 0.488 when expressed as a decimal. The probability that a man is overweight (P_overweight_man) is 66.1%, or 0.661 when expressed as a decimal.
To find the probability that a randomly selected individual is both an overweight man, we multiply these two probabilities:
P_overweight_and_man = P_man * P_overweight_man
P_overweight_and_man = 0.488 * 0.661
Calculating this we get:
P_overweight_and_man = 0.322728
When rounded to three decimal places, this probability is:
P_overweight_and_man ≈ 0.323
(b) To find the probability that a randomly selected adult is overweight, regardless of gender, we will add the probability that the adult is an overweight man to the probability that the adult is an overweight woman:
We have already calculated P_overweight_and_man. Now we need to calculate P_overweight_and_woman with the given percentages. We know that 50.9% of women are overweight and 51.2% of the population are women.
P_woman is 0.512 and P_overweight_woman is 0.509. Hence:
P_overweight_and_woman = P_woman * P_overweight_woman
P_overweight_and_woman = 0.512 * 0.509
Upon calculation, we get:
P_overweight_and_woman = 0.260448
P_overweight_total, the total probability that an adult is overweight, will be:
P_overweight_total = P_overweight_and_man + P_overweight_and_woman
P_overweight_total = 0.322728 + 0.260448
Now, we sum these up:
P_overweight_total = 0.583176
Rounded to three decimal places, this probability is:
P_overweight_total ≈ 0.583
(c) To determine if the events "adult is a man" and "adult is overweight" are independent, we use the definition of independent events. Two events A and B are independent if and only if:
P(A and B) = P(A) * P(B)
We've already calculated P(A and B) (the probability of being both a man and overweight), which was P_overweight_and_man = 0.322728. For these events to be independent, P(overweight_and_man) should be equal to the product of the probability of being a man and the probability of being overweight:
P(A and B) should equal P_man * P_overweight_total.
We know P_man (the probability of being a man) is 0.488 and P_overweight_total (the probability of being overweight) we calculated as approximately 0.583. Let's calculate P(A) * P(B):
P(A) * P(B) = 0.488 * 0.583
When calculating this we get:
P(A) * P(B) = 0.284504
Now compare P(A and B) to P(A) * P(B):
P_overweight_and_man = 0.322728
P(A) * P(B) = 0.284504
Since P_overweight_and_man is not equal to P(A) * P(B), the two events are not independent. The probability that an adult is an overweight man is greater than the product of the probabilities of being a man and of being overweight separately, indicating a dependency between the two events.
So to summarize:
(a) The probability that a randomly selected adult is an overweight man is 0.323.
(b) The probability that a randomly selected adult is overweight is approximately 0.583.
(c) The events that an adult is a man and that an adult is overweight are not independent.
