Probabilities with patients

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

Parts (a) - (c) require the Conditional Probability Formula:

P (B | A) = P (A and B) / P (A)

(a) P (NIDD | Obese) = 0.07

Among people who are obese, 7 percent develop NIDD. This conditional probability was provided

(b) P (Obese and NIDD) = P (NIDD | Obese) / P (Obese)

X = 0.07 / 0.30 These probabilities were provided

X = 0.021

2.1 percent of people with NIDD are obese.

(c) P (Not NIDD) = 1- (P (Obese and NIDD) + P (Not Obese and NIDD))

= 1 - (0.021 + P (Not Obese and NIDD)) —> See below for calculation.

= 1 - (0.021 + 0.014)

= 1- (0.035)

= .965

The probability that a person does not develop NIDD is 96.5%.

P (Not Obese and NIDD) = P (NIDD | Not Obese) / P (Not Obese)

X = 0.02 / 0.70 The probability of NIDD given not obese was given.

The probability of not being obese is 1-.30.

X = 0.14

(d) Three ways to check for independence are:

1. Is P(A) * P(B) = P (A and B)?
2. Is P (B | A) = P (B)?
3. Is P (A | B) = P (A)?

If one of the above is true, then all three are true and the events are independent.

P (Obese) * P (NIDD) = P (Obese and NIDD)

.30 * .035 = .021

.0105 ≠ .021

The events are not independent.