Cyan bars = positive contributions to E(X) · Red bars = negative contributions
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Probability distribution across all outcomes
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Law of Large Numbers — Live Simulation
Watch the running average converge to E(X) = — as the number of trials increases. The Law of Large Numbers guarantees this convergence.
Simulated Avg—
Theoretical E(X)—
Absolute Error—
Trials Run—
Set up a distribution in the Calculator tab, then click ▶ Run Simulation.
What is the Law of Large Numbers? No matter how random individual outcomes are, the long-run average of many trials will converge to the theoretical expected value E(X). This is why casinos always profit (negative E(X) per player), why insurance works (insurers can predict aggregate losses), and why index investing is effective over long horizons. The chart above shows this convergence in real time.
Common Real-World EVs
Scenario
E(X) per $1 wagered
Interpretation
Lottery ticket ($2)
≈ −$0.75
~75% house edge
Blackjack (basic strategy)
≈ −$0.005
0.5% house edge
Roulette (American, red bet)
≈ −$0.053
5.3% house edge
S&P 500 index (historical)
≈ +$0.10
~10% annual return
HYSA savings account
≈ +$0.045
~4.5% interest
How to Use This Calculator
1
Enter or Load Outcomes
Type your outcomes and probabilities, or click a preset chip (Fair Coin, Die, Lottery, etc.) to instantly load a distribution.
2
Check Probabilities Sum to 1
The probability sum indicator turns green when your distribution is valid. Adjust values if needed — decimals like 0.25 or fractions like 1/6 ≈ 0.1667.
3
Interpret Your Results
E(X) is the long-run average per trial. The contribution chart shows which outcomes drive E(X). Use the LLN tab to see convergence in action.
Formula & Methodology
Expected Value
E[X] = Σ(xᵢ × pᵢ)
The sum of each outcome multiplied by its probability gives the theoretical average outcome over many repetitions.
Variance
Var(X) = Σ(xᵢ−E[X])² × pᵢ
The expected squared deviation from the mean. High variance = high uncertainty even if E(X) is attractive.
Standard Deviation
σ = √Var(X)
The square root of variance. Expressed in the same units as the outcomes — easier to interpret than variance.
The weighted average of all possible outcomes, where weights are their probabilities. The long-run average if the experiment is repeated many times.
Outcome
A possible result of a random experiment — for example, the number on a die face or the profit from a business decision.
Probability P(x)
The likelihood of an outcome occurring. Must be between 0 and 1, and all probabilities must sum to exactly 1.
Variance Var(X)
A measure of spread. High variance means outcomes are scattered widely around E(X). Same E(X) with higher variance = more risk.
Fair Game
A situation where E(X) = 0: neither party has a long-run statistical advantage. Most casino games are not fair — they favor the house.
Real-World Examples
Example 1 — Fair Die
Six-sided Die Roll
x = [1, 2, 3, 4, 5, 6] p = [1/6 each ≈ 0.1667]
E[X] = 3.5 — the long-run average face value
Example 2 — Insurance
Home Fire Insurance
x = [+$99,000, −$1,000] p = [0.01, 0.99]
E[X] = 0. Insurer charges $1,000 for expected payout of $1,000. EV ≈ $0 for policyholder.
Example 3 — Business
Product Launch Decision
x = [+$100k, +$10k, −$50k] p = [0.4, 0.4, 0.2]
E[X] = +$34,000. Worth launching if development costs under $34k.
Expected Value: The Foundation of Rational Decision-Making
Why Expected Value Matters
Expected value is the cornerstone of decision theory, insurance pricing, gambling mathematics, and financial modeling. It answers a simple but powerful question: if you repeated this choice many times, what would you earn or lose on average? A positive E(X) suggests a favorable bet in the long run; a negative E(X) suggests a losing proposition. Casinos design every game to have a negative expected value for the player — which is why the house always wins over time.
Beyond the Average: Risk and Variance
Expected value alone doesn't capture the full picture. Two investments may share the same E(X) but have very different variances. A guaranteed $100 and a 50/50 chance of $0 or $200 both have E(X) = $100, but the latter is far riskier. In practice, rational decision-makers weigh both expected value and variance (or standard deviation) when evaluating options — this is the foundation of modern portfolio theory.
The Law of Large Numbers
The Law of Large Numbers (LLN) proves that as the number of trials grows, the sample average must converge to E(X). This is why insurance companies remain solvent despite paying large individual claims — the average payout over millions of policies is predictable. It's also why flipping a coin three times and getting three heads doesn't disprove that the coin is fair. Use the interactive LLN simulator in Tab 3 to see this convergence happen in real time.
Frequently Asked Questions
Yes. Casino bets, lottery tickets, and insurance policies typically have negative expected values for the buyer. A negative E(X) means you expect to lose money on average over many repetitions. Whether to participate anyway depends on your utility function and risk tolerance.
The simple average treats all outcomes equally. E(X) weights each outcome by its probability. For a fair die, both give 3.5. But for a biased die or a lottery, E(X) reflects the actual probabilities, not equal weighting.
Divide by 100. A 30% probability = 0.30. A 1-in-6 chance = 0.1667 (type 0.1667 for each face of a fair die). The probability sum indicator will show green when your entries total 1.0.
Variance measures risk. Two distributions with the same E(X) can have wildly different spreads. High variance means extreme outcomes (very good or very bad) are more likely. Low variance means results cluster near E(X). Investors often prefer lower variance for the same expected return.
A valid probability distribution must account for all possible outcomes. Probabilities summing to less than 1 would leave some probability "unaccounted for," and more than 1 is mathematically impossible. The calculator enforces this to ensure accurate results.
A game where E(X) = 0: neither player has a systematic long-run advantage. A fair coin flip for $1 is a fair game. Roulette is not fair — the house edge creates a negative E(X) for the player on every spin.
Yes. Expected value applies to any numerical outcome — number of goals in a match, points earned per turn in a board game, defects per production run, or years of life expectancy. The math is identical; only the interpretation changes.
The simulator samples from your distribution randomly, computing a running average after each trial. It then plots that running average vs. trial number alongside a dashed line for the theoretical E(X). As trials increase, you can see the running average zigzagging but steadily converging toward E(X).