expected-value

# Expected Value

## Table of Contents
- [Purpose](#purpose)
- [When to Use](#when-to-use)
- [What Is It?](#what-is-it)
- [Workflow](#workflow)
- [Common Patterns](#common-patterns)
- [Guardrails](#guardrails)
- [Quick Reference](#quick-reference)

## Purpose

Expected 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.

## When to Use

Use this skill when:

- **Investment decisions**: Should we invest in project A (high risk, high return) or project B (low risk, low return)?
- **Product bets**: Launch feature X (uncertain adoption) or focus on feature Y (safer bet)?
- **Resource allocation**: Which initiatives have highest expected return given limited budget?
- **Go/no-go decisions**: Is expected value of launching positive after accounting for probabilities of success/failure?
- **Pricing & negotiation**: What's expected value of accepting vs. rejecting an offer?
- **Insurance & hedging**: Should we buy insurance (guaranteed small loss) vs. risk large loss?
- **A/B test interpretation**: Which variant has higher expected conversion rate accounting for uncertainty?
- **Portfolio optimization**: Diversify to maximize expected return for given risk tolerance?

Trigger phrases: "expected value", "EV calculation", "risk-adjusted return", "probability-weighted outcomes", "decision tree", "should I take this gamble", "compare risky options"

## What Is It?

**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 s