4) Suppose that rate of Type 2 diabetes mellitus in 40-59 year olds is 7% among

Question

4) Suppose that rate of Type 2 diabetes mellitus in 40-59 year olds is 7% among Caucasians, 10% among African-Americans, 12% among Hispanics, and 5% among Asian-Americans. Suppose that the ethnic distribution of a city X among 40-59 year olds is 30% Caucasians, 25% African-Americans, 40% Hispanics, and 5% among Asian-Americans. What is the overall probability of Type 2 DB among 40-49 year olds in city X? Also, suppose a 40-59 year reports to your clinic with Type 2 DB. What is the probability that this person is African American? How about Asian American? (Hint: use generalized Bayes' Rule).

Explanation / Answer

P(Caucasians) = 0.3

P(African-Americans) = 0.25

P(Hispanics) = 0.4

P(Asian-American) = 0.05

P(Type 2 DB | Caucasians) = 0.07

P(Type 2 DB | African-American) = 0.1

P(Type 2 DB | Hispanics) = 0.12

P(Type 2 DB | Asian-American) = 0.05

a) P(Type 2 DB) = P(Type 2 DB | Caucasians) * P(Caucasians) + P(Type 2 DB | African-American) * P(African-American) + P(Type 2 DB | Hispanics) * P(Hispanics) + P(Type 2 DB | Asian-American) * P(Asian-American)

                           = 0.07 * 0.3 + 0.1 * 0.25 + 0.12 * 0.4 + 0.05 * 0.05

                           = 0.0965

b) P(African-American | type 2 DB) = P(Type 2 DB | African-American) * P(African-American) / P(type 2 DB)

                                                        = 0.1 * 0.25 / 0.0965

                                                        = 0.2591

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