Expected value and the shape of risk
Compare expected profits, the loss in a weak outcome, and the lowest cash balance before committing to an offer.
Compute expected outcomes for two choices and explain how downside size and cash constraints can change the preferred choice.
The choice at the food fair
Mara owns a catering business. A fair organizer offers two ways to serve Saturday lunch: earn a fixed fee for catering the staff meal, or run a stall and keep the sales. The stall looks more profitable on average. Mara still needs to know how much money she must put down on Friday and what happens if few people come.
Both businesses, amounts, and probabilities in this lesson are fictional. A calculator and paper are enough.
Recall Lesson 036: If the chance of strong attendance is 70% and attendance can be either strong or weak, what probability belongs on the weak branch? Why would this subtraction fail if there were a third possible outcome?
What expected value measures
A payoff is the result of a choice under one possible outcome. Here it means additional profit in dollars for one event, after all additional costs. A loss has a minus sign. Use the same measure and period for every choice.
Expected value is an average that gives each payoff a weight equal to its probability. Multiply each payoff by its probability, written as a decimal, then add the products. Include each possible outcome once. Their probabilities must total 1 because one of them will happen. OpenStax interprets this average in terms of repeated trials under the same conditions. OpenStax
For a single event, the average can be a dollar amount that Mara will never receive. It is useful for comparing offers. It does not tell her the most likely payoff, the largest modeled loss, or whether she can pay for ingredients before the fair opens.
Work the two offers
For the staff meal, Mara pays $600 on Friday and collects a guaranteed $1,500 on Saturday. For the stall, she pays $1,800 on Friday. Saturday receipts will be $4,200 with strong attendance or $600 with weak attendance. There are no other costs, taxes, payments, or effects on existing work. The organizer's 70% strong-attendance estimate is a planning assumption.
| Saturday outcome | Probability | Staff meal profit | Stall profit |
|---|---|---|---|
| Strong attendance | 70% | $900 | $4,200 − $1,800 = $2,400 |
| Weak attendance | 30% | $900 | $600 − $1,800 = −$1,200 |
Staff meal expected profit is (0.70 × $900) + (0.30 × $900) = $630 + $270 = $900. Attendance changes neither the contract payment nor its cost.
Stall expected profit is (0.70 × $2,400) + (0.30 × −$1,200) = $1,680 − $360 = $1,320. The stall leads by $420 on average. The $360 deduction is the loss weighted by its chance. On a weak Saturday Mara loses the full $1,200. On a strong one she earns $2,400. Neither day pays $1,320.
The shape of risk is visible in those two rows: the stall offers a larger gain and a 30% chance of a loss. If Mara can take only one booking, averaging the two offers together would answer no useful question. They are choices she controls; attendance is the uncertainty she faces after choosing.
Check the lowest cash balance
Mara has $2,500 cash before paying on Friday. She must leave at least $1,000 untouched at all times for other commitments. No borrowing or advance payment is available.
The staff meal leaves $2,500 − $600 = $1,900 on Friday. The stall leaves $2,500 − $1,800 = $700, breaching the reserve by $300 before a single customer arrives. Even its weak outcome would end Saturday with $700 + $600 = $1,300. Checking only the ending balance would miss Friday's shortfall.
Choose the staff meal under these terms. The $420 expected advantage tells Mara what makes the stall worth investigating. It does not finance the missing $300. If terms change, recheck both the timing of payments and the amount she could lose.
Exercise
Allow 8–10 minutes. You manage licensing at an established educational publisher. For one month's release, choose between a flat-fee license and a direct launch. Under the license, pay $400 today and collect a guaranteed $1,400 at month-end. Under the direct launch, pay $5,000 today and collect $8,000 if demand is strong or zero if it is weak.
Assume an 80% probability of strong demand and a 20% probability of weak demand. These are the only outcomes. All amounts are additional to existing business, all costs are paid today, and there are no other cash movements, costs, taxes, or financing. You have $6,500 available today and must maintain a $2,000 cash reserve. Among choices that meet the reserve in every outcome, management prefers higher expected profit.
- Calculate each choice's profit in both outcomes and its expected profit. How much can the publisher lose on the direct launch, and how likely is that loss?
- Calculate cash immediately after each upfront payment. Which plan can you approve under the reserve rule?
- Suppose starting cash were $8,500 instead. Check the direct launch's lowest cash balance and recommend a choice under the stated rule.
- With the larger cash balance, test a 75% chance of strong demand instead of 80%. What happens to the expected advantage? Name one piece of evidence you would seek before signing.
Try the decision before reading the answer.
Answer
The license earns $1,400 − $400 = $1,000 in either outcome, so its expected profit is $1,000. The direct launch earns $8,000 − $5,000 = $3,000 if strong and $0 − $5,000 = −$5,000 if weak.
Direct expected profit is (0.80 × $3,000) + (0.20 × −$5,000) = $2,400 − $1,000 = $1,400. It leads the license by $400. But the 20% loss branch costs $5,000 in full. Treating the weighted loss of $1,000 as the amount at risk would understate the downside by $4,000.
With $6,500 starting cash, the license leaves $6,100 today; the launch leaves $1,500. Approve the license. The launch falls $500 below the required reserve, even if strong sales eventually bring in $8,000.
With $8,500 starting cash, the launch leaves $3,500 today. In the weak outcome it still ends with $3,500; in the strong outcome it ends with $11,500. Both satisfy the $2,000 floor. Under management's rule, choose the direct launch for its higher expected profit. The additional cash changes what the publisher can afford. It changes neither the chance nor the size of the loss.
At 75% strong demand, direct expected profit becomes (0.75 × $3,000) + (0.25 × −$5,000) = $2,250 − $1,250 = $1,000. The expected advantage disappears. With equal expected profit, the license avoids a modeled $5,000 loss. Before favoring the launch, examine paid orders or results from comparable releases to see whether the 80% assumption is credible.
Recall answer: Weak attendance has probability 1 − 0.70 = 0.30, or 30%. With a third outcome, that 30% would cover all outcomes other than strong attendance; it could not all be assigned to weak attendance.
Applying this lesson
For each choice in your decision tree, label the outcome branches with their probabilities and payoffs. Alongside the average, record the largest modeled loss, its chance, and the lowest cash balance before collections.
Sources
Comments
Loading comments…