The chief loan officer of La Crosse Home Mortgage Company summarized the housing

Question

The chief loan officer of La Crosse Home Mortgage Company summarized the housing loans extended by the company in 2014 according to type and term of the loan. Her list shows that 70% of the loans were fixed-rate mortgages (F), 25% were adjustable-rate mortgages (A), and 5% belong to some other category (O) (mostly second trust-deed loans and home equity loans). Of the fixed-rate mortgages, 70% were 30-year loans, and 30% were 15-year loans; of the adjustable-rate mortgages, 30% were 30-year loans and 70% were 15-year loans; finally, of the other loans extended, 25% were 20-year loans, 60% were 10-year loans, and 15% were for a term of 5 years or less. Solve questions A through C below.

a) Select a tree diagram representing these data.

b) What is the probability that a home loan extended by La Crosse has an adjustable rate and is for a term of 15 years?

c) What is the probability that a home loan extended by La Crosse is for a term of 15 years?

Explanation / Answer

(a)

Of the fixed-rate mortgages (F), 70% were 30-year loans, and 30% were 15-year loans. So F should have two branches with 30 and 15 as root nodes. Of the adjustable-rate mortgages (A), 30% were 30-year loans and 70% were 15-year loans. So A should have two branches with 30 and 15 as root nodes.

Based on these criteria, top left (first) tree is the correct option.

b)

Probability that a home loan extended by La Crosse has an adjustable rate and is for a term of 15 years

= P(A) * P(15 year term | A) = 0.25 * 0.7 = 0.175

c)

By law of total probability,

Probability that a home loan extended by La Crosse is for a term of 15 years

= P(F) * P(15 year term | F) + P(A) * P(15 year term | A) + P(O) * P(15 year term | O)

= 0.7 * 0.3 +  0.25 * 0.7 + 0.05 * 0 = 0.385

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