Expected value applied to a business decision

Profit on every unit, a refund on some of them. Whether the company makes money, loses money or exactly breaks even is one subtraction — and ALEKS makes you commit to which of the three.

01 Worked Example Video

▸ VIDEO COMING SOON

A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

When they make money

A firm sells a device for $2,500, earning $250 profit. 8% are defective and get a full refund.

How much does the firm expect to make per device? Type a positive amount.

When they lose money

A firm sells an item for $1,200, earning $150 profit. 15% are defective and get a full refund.

How much does the firm expect to lose per item? Type a positive amount, as ALEKS does.

An insurance policy

An insurer charges a $530 annual premium. There is a 0.2% chance in any year of paying out $150,000.

What is the insurer’s expected profit per policy?

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

$E = \text{profit} - (\text{probability} \times \text{refund})$ earned on every unit, paid out on some

$E > 0$ make · $E < 0$ lose · $E = 0$ break even all three are live answers

The profit is collected on every unit sold; the refund goes out only on the defective ones. So the profit stands alone and the refund gets multiplied by its probability. Break-even is a real possibility here and not a trick — these questions are often built so the two figures cancel exactly.

Common slips

(1) Multiplied the profit by its probability too. The profit is earned on every unit, so it is not scaled. (2) Refunded the profit instead of the price. A full refund returns what the customer paid, which is the sale price, not the margin. (3) Picked “lose” and typed a negative number. The option already says lose, so the amount you type is positive.

04 Practice — Step by Step

Earned always, refunded sometimes

Digitalis makes a processor, the luteA, sold direct to the public for $3,600, making Digitalis a profit of $396.

A manufacturing flaw means some are defective and cannot be repaired. On those, Digitalis gives the customer a full refund. For each luteA there is an 11% chance it is defective and an 89% chance it is not.

If Digitalis sells many of these, should it expect to make money, lose money, or neither?

Step 1

The expected refund

11% of processors are refunded in full, at the $3,600 sale price. What is the expected refund per processor sold?

Step 2

The expected value

Digitalis earns $396 profit on every processor. Subtract the expected refund. What is the expected value per processor?

Step 3

So what is the answer?

The expected value is $0 per processor. Which statement should Digitalis choose?

Q12 Q14

Expected value applied to a business decision

What break-even actually means here