Question

Among many other pieces of information, the U.S. Census Bureau records the race or ethnicity of the residents of every county in every state. From these results, the bureau calculated a "diversity index" that measures the probability that two people chosen at random are of different races or ethnicities. Suppose that the census determined that in a county in Wisconsin 80% of its residents are white, 15% are black, and 5% are Asian. Calculate the diversity index for this county.

Answer 1

The diversity index is 0.335 or 33.5%

Explanation:

Probability index= probability that two people chosen at random are different ethnicity or races.

Let the probability of picking black = p(B) =0.15

probability of picking white= p(W)= 0.8

probability of picking Asian= p(A)=0.05

The probability of picking two people who are of different races or ethnicity can be gotten as:

Probability of picking: (White & Black) OR (White & Asian) OR (Black &Asian) OR (black &white) OR (Asian&white) OR (Asian& Black).

Using the notations given earlier, we can write the diversity index as:

Diversity index=p(W) x p(B)+ p(W) x p(A)+ p(B) x p(A) +p(B) x p(W)+ p(A) x p(W)+ p(A) x p(B)

Fixing the values for each term in the last expression gives:

Diversity index= 0.8x0.15 +0.8x0.05+ 0.15x0.05 +0.15x0.8+ 0.05x0.8+0.05x0.15

Diversity index= 0.12+0.04+0.0075+ 0.12+0.04+0.0075

Diversity index= 0.335 or 33.5%

Hence the diversity index is 0.335 or 33.5%