Question

The Scrub Professional Cleaning Service receives preliminary sales contracts from two sources: (1) its own agents and (2) building managers. Historically, 1/4 of the contracts have come from Scrub agents and 3/4 from building managers. Unfortunately, not all preliminary contracts result in actual sales contracts. Actually, only 3/8 of those preliminary contracts received from building managers result in a sale, whereas 7/8 of those received from Scrub agents result in sale. The net return (payoff) to Scrub from a sale is $1000. The cost of processing and following up on a preliminary contract that does not result in a sale is $150.

1. What is the probability that a preliminary contract leads to a sale?
2. What is the expected return (payoff) associated with a preliminary sales contract?
3. Which party (agents or building managers) contributes more to the expected return (payoff)?

Scrub keeps all of its sales filed by the source of reference; that is, it maintains one file for sales resulting from preliminary contracts submitted by Scrub agents and another for sales resulting from preliminary contracts submitted by building managers. Scrub knows that John Jones holds one of its sales contacts, and it wishes to have more information about him.

d. Which file should it search first to have the higher probability of finding his name? Hint: let S = preliminary contract results in sale; A = preliminary contract obtained by agent; M = preliminary contract by manager and use Bayes’ rule/theorem.

Answer 1

Given that, S = preliminary contract results in sale; A = preliminary contract obtained by agent; M = preliminary contract by manager

P(A) = ¼, P(M) = ¾, P(S/A) = 7/8, P(S/M) = 3/8, Net return(payoff) of sale =$1000 and cost=$150

Now we find

a. Probability that a preliminary contract leads to a sale = P(S) = P(A and S)+P(M and S)

= P(A)P(S/A)+P(M)P(S/M) = (1/4)(7/8)+(3/4)(3/8)=(7/32)+(9/32) =16/32 = ½ =½ 0.5

b. Expected Return(pay off) =

= Net return(payoff) of sale x Probability that a preliminary contract leads to a sale

= $1000x0.5 = $500

c. Expected Return(pay off) due to agent =

= Net return(payoff) of sale x Probability that a preliminary contract leads to a sale due to agent

= $1000xP(A and S) = $1000xP(A)P(S/A)= $1000x(1/4)(7/8)=$218.75

Expected Return(pay off) due to manager =

= Expected Return(pay off) - Expected Return(pay off) due to agent = $500 - $218.75 = $281.25

Therefore, Expected Return(pay off) due to manager is more than that of agent and hence, managers contribute more to expected return.

d. By Baye’s theorem P(A/S) = P(A)P(S/A)/ P(S) = (1/4)(7/8)/( ½) = 7/16 = 0.4375

P(M/S) = 1- P(A/S) = 1 – (7/16) = 11/16 = 0.6875

Therefore, the file for sales resulting from preliminary contracts submitted by building managers should be searched to have the higher probability of finding the name.

e.