Suppose that 3%
of all adults suffer from diabetes and that 29%
of all adults are obese. Suppose also that 2%
of all adults both are obese and suffer from diabetes.
(a)Find the probability that a randomly chosen adult is obese, given that he or she suffers from diabetes. Round your answer to the nearest hundredth.
(b)Find the probability that a randomly chosen adult who is obese also suffers from diabetes. Round your answer to the nearest hundredth.
To find the probability for these questions, we will use conditional probabilities.
(a) To find the probability that a randomly chosen adult is obese, given that he or she suffers from diabetes, we need to find the probability of being obese and having diabetes divided by the probability of having diabetes.
Let's denote the event "being obese" as O and the event "having diabetes" as D.
We are given the following probabilities:
- P(D) = 3% = 0.03 (probability of having diabetes)
- P(O) = 29% = 0.29 (probability of being obese)
- P(O and D) = 2% = 0.02 (probability of being obese and having diabetes)
The conditional probability can be calculated using the following formula:
P(O|D) = P(O and D) / P(D)
Substituting the values we have:
P(O|D) = 0.02 / 0.03 ≈ 0.67
Therefore, the probability that a randomly chosen adult is obese, given that he or she suffers from diabetes, is approximately 0.67.
(b) To find the probability that a randomly chosen adult who is obese also suffers from diabetes, we need to find the probability of being obese and having diabetes divided by the probability of being obese.
Using the same notation as before, we are given the following probabilities:
- P(D) = 3% = 0.03 (probability of having diabetes)
- P(O) = 29% = 0.29 (probability of being obese)
- P(O and D) = 2% = 0.02 (probability of being obese and having diabetes)
The conditional probability can be calculated using the following formula:
P(D|O) = P(O and D) / P(O)
Substituting the values we have:
P(D|O) = 0.02 / 0.29 ≈ 0.07
Therefore, the probability that a randomly chosen adult who is obese also suffers from diabetes is approximately 0.07.