Please, help. With explanations, please. 10% of all personal loans are in defaul
ID: 3202517 • Letter: P
Question
Please, help. With explanations, please.
10% of all personal loans are in default. Homeowners represent 20% of the loans in default and 70% of the loans that are not in default. Draw the probability tree. If a personal loan is selected at random, find the probability that it is to a homeowner and is not in default. If a personal loan is selected at random is to a homeowner, find the probability that the loan is in default. If a personal loan is selected at random, find the probability that the loan is not in default. Are the events "homeowner" and "loan in default" mutually exclusive? Are the events "homeowner" and "loan not in default" independent?
Explanation / Answer
let proability of loans to be default =P(D) =0.1
hence probabilty of loans not in default =P(ND) =1-P(D) =1-0.1 =0.9
probabilty it has been taken by home owner given it is default =P(H|D) =0.2
probabilty it has been taken by home owner given it is not in default =P(H|ND) =0.7
1)probability that it is to a homeowner and is not in default =P(HnND) =P(ND)*P(H|ND) =0.9*0.7 =0.63
probabilty that homeloan is to homeowner =P(H) =P(in default and belongs to home owner+not in default & belongs to home owner) =P(D)*P(H|D)+P(ND)*P(H|ND) =0.1*0.2+0.9*0.7=0.65
2)probability that the loan is in default given it is to a homeowner =P(D|H) =P(D)*P(H|D)/P(H) =0.1*0.2/0.65=0.0308
3)probability that the loan is not in default. =P(ND) =1-P(D) =1-0.1 =0.9
4)P(HnD) ==P(D)*P(H|D) =0.1*0.2 =0.02
and P(H)*P(D) =0.65*0.1=0.065
as P(HnD) is not equal to P(H)*P(D) . hence they are not independent.
5)P(HnND) =P(ND)*P(H|ND) =0.9*0.7=0.63
and P(H)*P(ND) =0.65*0.9 =0.585
as P(HnND) is not equal to P(H)*P(ND) . hence they are not independent.