# Conditional Probability of census burea

The question seems very straight forward , but when i attempted it the answer is wrong.

I just took the probability of all three groups and multiplied by themselves the intersections and addition them.

this is the question.

In early 2001, the United States Census Bureau started releasing the results of the latest census. Among many other pieces of information, the bureau recoded the race or ethnicity of the residents of every county in every state. From these results the bureau calculated a 'diversity index,' which measures the probability that two people chosen at random are of different races or ethnicities. The census determined that in a county in Wisconsin, 83% of its residents are white, 10% are black, and 7% are asian. Calculate the diversity index for this county.

Diversity Index =

What you're required to find is the following:$P(\text{Picking 2 people from different ethnicities})$. Let $C$ be the event that we pick 2 people and they are from 2 different ethnicities. $$P(\text{Picking 2 people from different ethnicities})=1-P(\text{2 chosen people belong to the same ethnicity})$$ The simpler question now is, given that you pick 2 people, what is the probability that they are from the same ethnicity; given that there are 3 different ethnicities spanning the whole Sample Space?