Expected Value

Builds onWeighted Mean (Portfolio Return) — if this page feels steep, start there.

E(X) = \sum_{i=1}^{n} p_i X_i

$E(X)$: the expected value of X — read 'E of X': the probability-weighted average outcome
$p_i$: the probability of scenario i, as a decimal (30% is 0.3)
$X_i$: the payoff if scenario i happens
$\sum p_i X_i$: multiply each payoff by its own probability, then add — likelier outcomes count for more

Why it must be true

The expected value is a probability-weighted average: each outcome contributes in proportion to how likely it is. It is what your average result would converge to if the gamble were repeated many times.

Note what it is not: the most likely outcome, and often not even a possible outcome — a coin flip paying $0 or $100 has an expected value of $50, which never occurs. It is the fair price of the gamble, not a prediction of any single run.

The derivation

Imagine the gamble run $N$ times for large $N$ . Outcome $X_i$ occurs about $p_i N$ times, so total winnings are $\sum_i (p_i N) X_i$ . The average per run:

E(X) = \frac{\sum_i p_i N X_i}{N} = \sum_i p_i X_i

The weights must exhaust all possibilities: $\sum p_i = 1$ .

When to reach for it

Discrete scenarios with probabilities attached — expected P&L, scenario-weighted forecasts, fair value of a gamble.

Listen for

“with probability … pays …”“boom / base / recession scenarios”“expected payoff / expected cash flow”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Anchor on the most likely outcome, then drift toward the others: each alternative pulls the answer by (its probability) × (its distance from the anchor).
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Two outcomes are instant: a 30/70 split between \ pitfalls: [00 and $200 is 0.3 × 100 + 0.7 × 200 = \ pitfalls: [70. The answer always leans toward the likelier payoff.
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Probability-weighted answers rarely equal any single scenario — a choice exactly matching one given payoff is usually the 'picked the mode' trap.

Traps in applying it

✗Probabilities that don't sum to 1.
✗Simple-averaging the outcomes as if scenarios were equally likely.
✗Presenting the EV as the likely result — it may be an impossible outcome (no coin pays its expected value).

Limits & criticisms

Expected value is risk-blind: a certain $1M and a coin flip for $2M have the same EV but are not the same offer, which is why utility theory exists. The scenario probabilities are usually subjective estimates dressed as data, and for heavy-tailed payoffs the EV alone can be actively misleading (the St. Petersburg paradox).

Where it came from

Probability theory was invented to price a gamble. In 1654 Blaise Pascal and Pierre de Fermat exchanged letters on the "problem of points" — how to split the pot of an interrupted game fairly — and their answer, weighting each outcome by its probability, is expected value. Christiaan Huygens published the first treatise on it in 1657.

Today it is the spine of decision-making under uncertainty: scenario-weighted forecasts, insurance pricing, and (under risk-neutral probabilities) the valuation of every derivative.

Where it leads

Master this and the following come almost for free:

Total Probability Rule One-Period Binomial Option Value

Drill this formula Back to the library