Expected Value Calculator

**Expected value (discrete):** E(X) = Σ x × P(x) Sum of each outcome times its probability. **Worked example: dice roll** Outcomes: 1, 2, 3, 4, 5, 6. Each P = 1/6. E(X) = 1×(1/6) + 2×(1/6) + 3×(1/6) + 4×(1/6) + 5×(1/6) + 6×(1/6) = 3.5 So average roll of 1 die is 3.5. **Worked example: lottery** Win $1,000,000 with P = 1/300,000,000. Cost: $2 ticket. EV = 1,000,000 × (1/300,000,000) - 2 × 1 = 0.0033 - 2 = -$1.997 Each ticket has expected loss of ~$2 (slight underestimate due to other prizes). **Variance:** Var(X) = E[(X - E(X))²] = Σ (x - μ)² × P(x) **Standard deviation:** SD(X) = √Var(X) **Properties of expected value:** 1. **Linearity**: E(aX + b) = aE(X) + b 2. **Sum**: E(X + Y) = E(X) + E(Y) (always) 3. **Product (independent)**: E(XY) = E(X)E(Y) only if X and Y independent 4. **Constant**: E(c) = c **Common expected value calculations:** **Casino games:** | Game | House edge | Player EV per $100 bet | |---|---|---| | Blackjack (basic strategy) | 0.5% | -$0.50 | | Baccarat (banker) | 1.06% | -$1.06 | | Craps (pass line) | 1.4% | -$1.40 | | Roulette (single 0) | 2.7% | -$2.70 | | Roulette (double 0) | 5.3% | -$5.30 | | Slot machines | 5-15% | -$5 to -$15 | | Keno | 25% | -$25 | | Lottery | 50% | -$50 | All negative EV for player — casino edge built in. **Insurance:** Premium = Expected claim + Operating costs + Profit margin For policyholder: EV = (P × policy payout - premium) < 0. For company: EV = premium - P × policy payout > 0. Insurance has negative EV for individual but reduces variance (peace of mind). **Investment:** E(return) = Σ (probability × return) For long-term: stock market expected ~7% annual real return. Variance is high (volatility); but compounded over time, EV dominates. **Decision theory:** Choose option with highest EV when: - Risk-neutral (don't care about variance). - Many repeated decisions. - Outcomes are not catastrophic. Don't use EV alone when: - Risk-averse (variance matters). - Single critical decision. - Catastrophic outcomes possible. - Small samples. **St. Petersburg paradox:** Coin flip game: heads on first flip = $2; if tails, continue. Each successive heads doubles ($4, $8, $16, ...). E(payout) = $2 × 0.5 + $4 × 0.25 + $8 × 0.125 + ... = $1 + $1 + $1 + ... = infinity! But how much would you pay to play? Probably $10-20. Shows EV alone isn't sufficient for decisions. **Allais paradox:** Most people prefer: - Option A: 100% chance of $1M. - Over Option B: 89% of $1M + 10% of $5M + 1% of $0. But EV(B) > EV(A). Demonstrates that risk preferences violate strict EV maximization. **Practical interpretation:** - EV positive: gain expected in long run. - EV negative: loss expected in long run. - EV = 0: break-even. - Magnitude matters: bigger EV = more favorable. **Common mistakes:** - Confusing EV with "most likely" outcome. - Ignoring variance and risk. - Treating one-time decisions like repeated. - Forgetting to include all possible outcomes. **Use case examples:** **Bet evaluation:** EV of $5 bet on coin flip (50% win $5, 50% lose $5) = 0. Break-even. Wouldn't take if I value $5 differently win vs lose. **Insurance:** $300/year auto insurance vs P(crash) = 0.01 × $30,000 = $300 EV. Break-even in pure EV terms. But variance reduction valuable. **Stocks:** Mean return 8%, SD 16%. EV positive. But large variance means individual year may lose 20%. **Conditional expected value:** E(X | event) = expected value given event happens. Useful for partial information problems. **Long-run interpretation:** Per the Law of Large Numbers, sample average → expected value as n → ∞. Over many trials, average outcome converges to EV.