The Law of Large Numbers and Expected Value

The one idea: Expected value is a weighted average of every possible outcome, weighted by how likely each one is — and the Law of Large Numbers says that as you repeat a random process more and more times, the average of your actual results converges toward that expected value. It says nothing at all about what happens on any one trial; it's a statement about long-run averages, full stop.

⚠️ The trap: "Roulette has landed red six times in a row, so black is due — the law of averages will balance it out." That's the gambler's fallacy, and it inverts what the Law of Large Numbers actually says. **The law describes what happens to the average over many future trials, not a correction mechanism that owes you a black to cancel out past reds — the wheel has no memory, each spin is independent, and the probability of black on the next spin is exactly what it was on the first, unmoved by the streak.**

Expected value: a weighted average of outcomes

For a discrete random variable X that can take values x with probabilities f(x), the expected value is:

E(x) = μ = Σ x · f(x)

— multiply each possible outcome by its probability, then add them up. It's a weighted average: outcomes that are more likely count for more, exactly the way a heavier item pulls a see-saw's balance point toward it.

Worked example. A car dealership, DiCarlo Motors, tracks daily sales and finds this probability distribution for automobiles sold in a day:

| Automobiles sold (x) | 0 | 1 | 2…