Compare uncertain strategies and consequences

Try

A fictional event organizer must choose between two supply plans. Plan A loses 6 budget units regardless of the weather. Plan B loses 2 units in ordinary weather and 14 units during a disruption. These are teaching numbers, not prices or advice for a real event. Lower loss is better. Before calculating, explain which plan has the smaller worst listed loss and which might have the smaller average loss.

Let p mean the probability of disruption, between 0 and 1. The ordinary-weather probability is 1 − p. The plans face the same weather state; choosing a plan does not alter that probability in this model. If a plan changed the chance of disruption, the table and calculation would need to reflect that causal change.

Predict

At p = 1/4, predict which plan has lower expected loss. Then predict whether the same plan remains preferable by that criterion when p = 1/2. State your criterion explicitly: “lower expected loss” is a mathematical comparison, while “better plan” can also depend on affordability, risk tolerance and omitted consequences.

Plan	Ordinary weather	Disruption	Largest listed loss
A	6	6	6
B	2	14	14

At p = 1/4, Plan B’s weighted mean is (3/4)2 + (1/4)14 = 5 units, compared with 6 for A. A one-unit expected saving does not promise a one-unit saving at this particular event. Its realized loss is either 2 or 14.

Experiment

The live playground deliberately uses different consequences: the variable strategy loses 2 or 18 units, and the guaranteed loss can be changed. Predict the crossover probability before revealing the calculations. Vary the probability while holding both loss columns fixed, then vary the guaranteed amount while holding the probability fixed. Separate those experiments in your notes.

Compare loss, probability and risk

Fictional loss units: one strategy loses a guaranteed amount in either state; the other loses 2 in a normal state and 18 in a disruption. Predict the lower expected loss and whether that choice also has the smaller worst listed loss.

Disruption probability p00.1250.250.3750.50.751Guaranteed loss4681012

Reveal calculationsReset explorer

Commit to a prediction, then reveal the table and fixed-scale plot. Change one control and explain the difference.

All settings and traces are temporary. This explorer does not save an answer or award mastery; keep your assumptions and reasoning in the written practice.

Find a setting where the strategies tie in expected loss but differ in variability. Record the table, expected losses, and largest listed losses. Explain why the zero-to-twenty vertical axis remains fixed as settings change. The graph compares like quantities in the same units; it does not assign a general score to the quality of a decision.

Understand

For the static worked case, the expected loss of B is 2(1 − p) + 14p = 2 + 12p. Plan A’s expected loss is always 6. Equality occurs when 2 + 12p = 6, giving p = 1/3. Below that threshold, B has smaller expected loss; above it, A does. At the threshold the expected-loss criterion alone cannot distinguish them.

This calculation is a sensitivity analysis: it identifies how the conclusion changes with an uncertain input. If the plausible range for p spans one third, declaring one strategy universally preferable by expected loss hides information. A useful report can instead show the threshold and explain what additional evidence about p could resolve the comparison. A value of p outside 0 to 1 is not a valid probability and cannot be repaired by simply plugging it into the formula.

At p = 1/4, B has variance (3/4)(2 − 5)² + (1/4)(14 − 5)² = 27 squared budget units. A has variance zero. Variance describes spread around the mean; the largest listed loss describes a different aspect of risk. Neither automatically encodes the organizer’s constraints. If losing 14 units would make the event infeasible, the one-unit expected saving may be unacceptable despite correct arithmetic.

A payoff formulation reverses the optimization direction. If payoff is negative loss, maximizing expected payoff gives the same ordering as minimizing expected loss. If a table lists prizes before fees, subtract fees before comparing net payoffs. Keep the state probabilities, units and consequence definitions visible. Comparing one plan’s gross prize to another’s net profit would not be a valid expected-value comparison.

Apply

Write two defensible recommendations for the worked case at p = 1/4: one for a decision maker whose stated goal is minimum expected loss and who can absorb 14 units, and one for a decision maker with a strict maximum acceptable loss of 8 units. Both can use the same probability table and reach different choices because the constraints differ. Do not invent a numerical utility score for their preferences.

Then critique the model. Are ordinary weather and disruption the only relevant states? How was p estimated? Are setup costs included in both plans? Would the loss amounts vary within a state? Name one additional observation that would matter for the decision and explain which input or assumption it would change. These questions connect the calculation to evidence without pretending that a small synthetic model settles a real business choice.

Continue

The practice asks for both calculations and a justified interpretation. Retain the probability table, threshold, risk comparison and assumptions as separate pieces of evidence. The next lesson asks a different design question: can a random procedure give several people equal chances, and what happens if a bounded procedure cannot make a decision?