Question

Over the summer of 2004, we took a look at the graduate students at a major university. About 85.57% of them were Asian. Approximately 64% of the Asian graduate students lived in Family Housing apartments, while 7.05% of the non-Asian graduate students lived in Family Housing apartments. What percentage of graduate students living in Family housing apartments were Asian? (Provide answer in the form 78.23 if 78.23% for example.)

Answer 1

We are given here that:
P( Asian) = 0.8557,
P( Family Housing apartments | Asian) = 0.64
P( Family Housing apartments | Non Asian) = 0.0705

Using law of total probability, we have here:
P( Family Housing apartments ) = P( Family Housing apartments | Asian) P(Asian) + P( Family Housing apartments | Non Asian)P(Non Asian)

= 0.64*0.8557 + 0.0705*(1 - 0.8557)

= 0.55782115

Now using Bayes theorem, we have here:
P( Asian | Family Housing apartments) = P( Family Housing apartments | Asian) P(Asian) / P( Family Housing apartments )

= 0.64*0.8557 / 0.55782115

= 0.9818

Therefore 0.9818 is the required conditional probability here.