Statistics- probability

Question

In studies linking obesity to Non-Insulin Dependent Diabetes (NIDD) it is thought that obesity genes interact with diabetes susceptibility genes. In industrialized countries 30% of the population is estimated to be obese. Among people who are obese, seven percent develop NIDD; while only 2% of non-obese people develop NIDD.

(a) What is the probability that a person is obese and develops NIDD?

(b) What is the probability that a NIDD person is obese?

(c) What is the probability that a person does not develop NIDD?

(d) Are the two events “Obese” and “NIDD” statistically independent? Clearly prove your answer.

Answer 1

Use O for Obese, and D for Develops Diabetes.

Given:

P(O) = .3

P(O^c) = 1-.3 = .7 (not obese)

P(D|O) = .07 (translated: given someone is obese, what is the probability that they develop diabetes)

P(D|O^c) = .02 (translated: given someone is not obese, what is the probability that they develop diabetes)

We will be using Bayes Theorem which says in general P(A|B) = P(A∩B) / P(B)

Now we can find P(D).

P(D) = P(D|O)*P(O) + P(D|O^c)*P(O^c)

P(D) = .07*.3 + .02*.7 = .035

a. P(O∩D) = P(O) * P(D|O) = .3*.07 = .021

b. P(O|D) = P(O∩D) / P(D) = .021 / .035 = 0.6

c. P(D^c) = 1-P(D) = .965

d. Independent if P(O∩D) = P(O) * P(D)

.021 ≠ .3*.035, so not independent