Question

Part A:

The aggregate percent chance of a customer buying is 35%. Customers are divided into two segments: 20% are “Big Spenders” who are worth $11,000 in profit to the dealership if they buy; they have a 15% chance of buying. The remainder of the customers are “Ordinary Joes,” who are worth $3,000 if they buy. What is the expected value of the next customer who walks in, assuming that it is completely random who walks in and whether they buy is also random and independent of their decision to walk in?

Part B:

Assume that the information in the above was wrong, and that Big Spenders actually purchase only 12% of the time (all other information is accurate). Salespeople get a 25% commission on profit, and they fight incessantly over Big Spenders when they walk in. Is this reasonable? What is the actual expected difference in commission to a salesperson for a Big Spender vs. an Ordinary Joe? Qualitatively (or quantitatively) which segment has greater risk?

Answer 1

A) Let B denotes that the customer is a big spender, while O denotes that the customer is Ordinary Joe. Given that there are 20% B and 80% O. Also we know that the chances of a customer buying are 0.35. Given that 15% of the B buy. Hence we'll find the chances of O buying. Let p be the probability of O buying then it satisfies:

0.2*0.15 + 0.8*p = 0.35 => 0.8*p = 0.35 - 0.03 => p = 0.4.

Thus chances of O buying are 0.4. Now given that profit associated with B is 11,000 while that with Ois 3,000. Thus expected profit from a random customer is:

0.2*0.15*11,000 + 0.8*0.4*3,000 = $1290

B) Here given that 12% of the B buy. Hence we'll find the chances of O buying. Let p be the probability of O buying then it satisfies: 0.2*0.12 + 0.8*p = 0.35 => 0.8*p = 0.35 - 0.024 => p = 0.4075

Thus chances of O buying are 0.4075. Now we'll calculate the profit associated with B and O separately.

Profit associated with a random B is 0.12*11000 = 1320, while the profit associated with a random O is 0.4075*3000 = 1222.5

Here associated profit with B is more than O. Thus it's reasonable to fight incessantly over Big Spenders when they walk in.

The commission associated is given to be 25% thus commission associated with B is 0.25*1320 = $264, while the commission associated with O is 0.25*1222.5 = $305.625