A realtor knows that 15% of listed houses are sold while the rest are delisted.

Question

A realtor knows that 15% of listed houses are sold while the rest are delisted. He does some research and finds the selling price of a house has a mean of $600(k) and standard deviation of $150(k). Prices of individual houses are independent of each other, and whether a house is sold or not is independent for different houses. Suppose the realtor has 80 houses to sell, and he makes 5% commission on each house sold.

(a) What is the probability that the realtor sells more than 10 houses?

(b) Suppose that the realtor has sold 36 houses. What is the probability that he makes more than $1400(k)? State any assumption you need to solve this question. Explain briefly why this assumption is needed. If no assumption is needed, please do state so.

(c) Suppose that the realtor has sold 12 houses. What is the probability that he makes more than $400(k)? State any assumption you need to solve this question. Explain briefly why this assumption is needed. If no assumption is needed, please do state so.

Explanation / Answer

(a) What is the probability that the realtor sells more than 10 houses?

mean=n*p=80*0.15 =12

standard deviation =sqrt(n*p*(1-p))=sqrt(80*0.15*0.85) =3.193744

So P(X>10) = P((X-mean)/s >(10-12)/3.193744)

=P(Z>-0.63) =0.7357 (from standard normal table)

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(b) Suppose that the realtor has sold 36 houses. What is the probability that he makes more than $1400(k)? State any assumption you need to solve this question. Explain briefly why this assumption is needed. If no assumption is needed, please do state so.

We need to assume that the population follows Normal distribution because we can use standard normal table to do the calculation.

mean=600*0.05*36 = 1080

standard deviation =0.05*150*36 =270

So P(X>1400) = P((X-mean)/s >(1400-1080)/270)

=P(Z>1.19) =0.1170 (from standard normal table)

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(c) Suppose that the realtor has sold 12 houses. What is the probability that he makes more than $400(k)? State any assumption you need to solve this question. Explain briefly why this assumption is needed. If no assumption is needed, please do state so.

We need to assume that the population follows Normal distribution because we can use standard normal table to do the calculation.

mean=600*0.05*12 = 360

standard deviation =0.05*150*12 =90

So P(X>400) = P((X-mean)/s >(400-360)/90)

=P(Z>0.44) =0.3300 (from standard normal table)

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