Use when making decisions under uncertainty with quantifiable outcomes, comparing risky options (investments, product bets, strategic choices), prioritizing projects by expected return, assessing whether to take a gamble, or when user mentions expected value, EV calculation, risk-adjusted return, probability-weighted outcomes, decision tree, or needs to choose between uncertain alternatives.
Calculates probability-weighted expected value to compare risky alternatives under uncertainty. Use when user mentions "expected value," "EV calculation," or needs to choose between uncertain options like investments, product bets, or strategic decisions.
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resources/evaluators/rubric_expected_value.jsonresources/methodology.mdresources/template.mdExpected Value (EV) provides a framework for making rational decisions under uncertainty by calculating the probability-weighted average of all possible outcomes. This skill guides you through identifying scenarios, estimating probabilities and payoffs, computing expected values, and interpreting results while accounting for risk preferences and real-world constraints.
Use this skill when:
Trigger phrases: "expected value", "EV calculation", "risk-adjusted return", "probability-weighted outcomes", "decision tree", "should I take this gamble", "compare risky options"
Expected Value (EV) = Σ (Probability of outcome × Value of outcome)
For each possible outcome, multiply its probability by its value (payoff), then sum across all outcomes.
Core formula:
EV = (p₁ × v₁) + (p₂ × v₂) + ... + (pₙ × vₙ)
where:
- p₁, p₂, ..., pₙ are probabilities of each outcome (must sum to 1.0)
- v₁, v₂, ..., vₙ are values (payoffs) of each outcome
Quick example:
Scenario: Launch new product feature. Estimate 60% chance of success ($100k revenue), 40% chance of failure (-$20k sunk cost).
Calculation:
Interpretation: On average, launching this feature yields $52k. Positive EV → launch is rational choice (if risk tolerance allows).
Core benefits:
Copy this checklist and track your progress:
Expected Value Analysis Progress:
- [ ] Step 1: Define decision and alternatives
- [ ] Step 2: Identify possible outcomes
- [ ] Step 3: Estimate probabilities
- [ ] Step 4: Estimate payoffs (values)
- [ ] Step 5: Calculate expected values
- [ ] Step 6: Interpret and adjust for risk preferences
Step 1: Define decision and alternatives
What decision are you making? What are the mutually exclusive options? See resources/template.md.
Step 2: Identify possible outcomes
For each alternative, what could happen? List scenarios from best case to worst case. See resources/template.md.
Step 3: Estimate probabilities
What's the probability of each outcome? Use base rates, reference classes, expert judgment, data. See resources/methodology.md.
Step 4: Estimate payoffs (values)
What's the value (gain or loss) of each outcome? Quantify in dollars, time, utility. See resources/methodology.md.
Step 5: Calculate expected values
Multiply probabilities by payoffs, sum across outcomes for each alternative. See resources/template.md.
Step 6: Interpret and adjust for risk preferences
Choose option with highest EV? Or adjust for risk aversion, non-monetary factors, strategic value. See resources/methodology.md.
Validate using resources/evaluators/rubric_expected_value.json. Minimum standard: Average score ≥ 3.5.
Pattern 1: Investment Decision (Discrete Outcomes)
Pattern 2: Portfolio Allocation (Multiple Options)
Pattern 3: Sequential Decision (Decision Tree)
Pattern 4: Continuous Distribution (Monte Carlo)
Pattern 5: Competitive Game (Payoff Matrix)
Critical requirements:
Probabilities must sum to 1.0: If you list outcomes, their probabilities must be exhaustive (cover all possibilities) and mutually exclusive (no overlap). Check: p₁ + p₂ + ... + pₙ = 1.0.
Don't use EV for one-shot, high-stakes decisions without risk adjustment: EV is long-run average. For rare, irreversible decisions (bet life savings, critical surgery), consider risk aversion. A 1% chance of $1B (EV = $10M) doesn't mean you should bet your house.
Quantify uncertainty, don't hide it: Probabilities and payoffs are estimates, often uncertain. Use ranges (optimistic/pessimistic), sensitivity analysis, or distributions. Don't pretend false precision.
Consider non-monetary value: EV in dollars is convenient, but some outcomes have utility not captured by money (reputation, learning, optionality, morale). Convert to common scale or use multi-attribute utility.
Probabilities must be calibrated: Don't use gut-feel probabilities without grounding. Use base rates, reference classes, data, expert forecasts. Test: are your "70% confident" predictions right 70% of the time?
Account for correlated outcomes: If outcomes aren't independent (economic downturn affects all portfolio companies), correlation reduces diversification benefit. Model dependencies.
Time value of money: Payoffs at different times aren't equivalent. Discount future cash flows to present value (NPV = Σ CF_t / (1+r)^t). EV should use NPV, not nominal values.
Stopping rules and option value: In sequential decisions, fold-back induction finds optimal strategy. Don't ignore option to stop early, pivot, or wait for more information.
Common pitfalls:
Key formulas:
Expected Value: EV = Σ (pᵢ × vᵢ) where p = probability, v = value
Expected Utility (for risk aversion): EU = Σ (pᵢ × U(vᵢ)) where U = utility function
Net Present Value: NPV = Σ (CF_t / (1+r)^t) where CF = cash flow, r = discount rate, t = time period
Variance (risk measure): Var = Σ (pᵢ × (vᵢ - EV)²)
Standard Deviation: σ = √Var
Coefficient of Variation (risk/return ratio): CV = σ / EV (lower = better risk-adjusted return)
Breakeven probability: p* where EV = 0. Solve: p* × v_success + (1-p*) × v_failure = 0.
Decision rules:
Sensitivity analysis questions:
Key resources:
Inputs required:
Outputs produced:
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