E(X)Expected Value Calculator
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Expected Value (EV), also called mathematical expectation, is the probability-weighted average of all possible outcomes of a random variable. It represents the long-run average result if an experiment is repeated many times. EV is the cornerstone of decision theory, financial modelling, insurance, and gambling strategy.
- 1E(X) = Σ xᵢ × P(xᵢ) for all outcomes i
- 2All probabilities must sum to 1
- 3Variance σ² = Σ P(xᵢ) × (xᵢ − E(X))²
- 4Standard deviation σ = √Variance — measures spread around the mean
- 5A positive EV game is profitable in the long run; negative EV is a loss
Roll a fair die — win £6 if you roll 6, lose £1 otherwise=EV = 6×(1/6) + (−1)×(5/6) = +£0.17/rollPositive EV — play!
UK lottery — £2 ticket, jackpot £5M, odds 1 in 45M=EV ≈ −£1.89 per ticketVery negative EV
| Scenario | EV | Interpretation |
|---|---|---|
| Fair coin flip £1 vs £1 | £0.00 | No edge — fair game |
| Roulette (red/black, UK) | −£0.027 per £1 | 2.7% house edge |
| Blackjack (optimal strategy) | −£0.005 to −£0.01 | Lowest casino house edge |
| S&P 500 (historical real return) | +~6.8% pa | Long-run positive EV |
| Insurance premium | Negative EV financially | But positive utility (risk transfer) |
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