Business Statistics! Please help

Question

You are selling a product in an area where 30 % of the people live in the city and the rest live in the suburbs. Currently 20 % of the city dwellers user your product and 10 % of the suburbanites use your product.  Given that someone uses your product, what is the probability that the person lives in the city?

What do I know?  

a) What do I want to find out?

b) What do I expect the answer to be? 

c) How do I go from what I know to what I want to find?

d) Is the answer consistent with what I expected?  

Answer 1

This is using Bayes Principle which says that P(B)*P(B|A) = P(A)P(A|B) = P(A or B)

P(C) = .3 (Prob of living in City)

P(S) = 1-.3 = .7 (Prob of living in Suburb)

P(P|C) = .2 (Prob of user of product if the person lives in the city)

P(P|S) = .1 (Prob of user of product if the person lives in the superb)

We also know the probabilities of not being a user given C or S

We are looking for P(C|P) which from the equation is P(P|C)P(C) / P(P)

Wait, we need to calculate the P(P), but that's just the sum of the different user types:

P(P|C)P(C) + P(P|S)P(S) = (.2)(.3) + (.1)(.7) = .13

so, finally, (.2)(.3)/(.13) = .46

It makes sense that of the population of people who own the product, about half are in the city. They are twice as likely to own the product as people in the suburbs, but the populations are about 2:1 the other way, so the odds are even once you know the person is an owner.

Please consider tutoring.

PS. For simple problems (rather than going through the Bayes hoohah, I just find the weighted average of the probabilities of C and S weighted with being the probability of being a product owner given where they live (that's the .13 calculation) and divide it into the term that is the probability of just the City owners which is the first term in the .13 calculation. If that's just confusing - ignore it.