Expected Value
Should you spend $50,000 on a campaign that will probably flop — but might be huge? Expected value gives you a single number to make that call, and teaches you exactly when to trust it.
What you'll learn
- Expected value: the average payoff across many repetitions of the same decision
- How to compute EV as a weighted sum of outcomes
- Why a likely-to-fail bet can still be a smart bet
- The law of large numbers and why it limits EV thinking to repeatable decisions
- When positive EV is NOT enough — variance, ruin risk, and survivable bets
Before you start
Here is a decision that every marketing team, product manager, and investor faces repeatedly: the numbers look uncertain, the likely outcome is disappointing, and yet something about the upside feels compelling. How do you evaluate it without either panicking or fooling yourself?
The tool is expected value — the average payoff you would get if you could repeat the same decision many times under identical conditions. It does not predict what will happen this one time. It tells you whether the bet is structurally good or structurally bad, averaged across many such bets.
Two building blocks first
To compute expected value you need two ingredients.
Probability (the chance something happens) is a number between 0 and 1, where 0 means impossible and 1 means certain. A 40% chance is written as 0.40. It represents how often an outcome would occur if you repeated the situation a very large number of times.
Payoff (what you actually receive or lose if that outcome happens) is a dollar amount — or any numeric measure of value. Payoffs can be positive (you gain) or negative (you lose).
Expected value (EV) = the sum, over every possible outcome, of (probability of that outcome times its payoff)
In plain text:
EV = (probability of outcome A × payoff of A)
+ (probability of outcome B × payoff of B)
+ ... (one term per outcome)All probabilities must add up to 1 — something has to happen.
The campaign example, step by step
Your marketing team proposes a $50,000 campaign. The analysis surfaces two scenarios:
| Outcome | Probability | Extra gross profit generated |
|---|---|---|
| Campaign works | 40% | $200,000 |
| Campaign flops | 60% | $0 |
Step 1 — EV of the gross profit outcome (before subtracting the cost):
EV of outcome = 0.40 × $200,000 + 0.60 × $0
= $80,000 + $0
= $80,000If you ran this exact campaign one hundred times, on average you would generate $80,000 in gross profit per run.
Step 2 — Net EV (after subtracting the $50,000 cost you pay regardless):
Net EV = EV of outcome − cost
= $80,000 − $50,000
= +$30,000A net EV of +$30,000 means this is a positive-EV decision. On average, across many similar campaigns, you come out $30,000 ahead per campaign — even though the campaign fails 60% of the time.
The core intuition
The result surprises many people: a decision can be smart even when the most likely single outcome is failure, as long as the rare win is large enough to more than compensate.
Think of it this way. Imagine you run this style of campaign ten times over five years. You expect it to flop 6 of those 10 times ($0 each) and succeed 4 of those 10 times ($200,000 gross profit each). Total gross profit across ten campaigns: 4 times $200,000 = $800,000. Total cost: 10 times $50,000 = $500,000. Net gain over ten campaigns: $300,000 — or $30,000 per campaign on average. That is exactly what the net EV formula predicted.
You are not betting that this campaign will work. You are betting that campaigns of this profile are worth running as a practice. That is a different, and often more defensible, claim.
The diagram
A chance node fans into two outcomes. The expected gross-profit EV is $80,000; subtract the $50,000 cost and net EV is +$30,000.
The crucial caveat — when to stop trusting EV
Expected value rests on a concept called the law of large numbers (the mathematical fact that an average converges to its expected value as you take more and more samples). The catch: you need many repetitions for that averaging to happen.
If you run fifty marketing campaigns a year of similar character, EV is an excellent guide — over time, your portfolio of bets will land near the expected payoff.
If this is a one-shot, bet-the-company decision — you can only make it once and a loss would threaten the business — EV alone is not enough. A 60% chance of losing $50,000 in a world where your cash runway is $60,000 is very different from a 60% chance of losing $50,000 when you have $5 million in reserve. The average is the same; the existential risk is not.
For one-shot decisions you must also weigh variance (how spread out the outcomes are) and ruin risk (the probability of an outcome bad enough to end the game). EV is necessary but not sufficient.
Putting it together — a decision checklist
When someone presents a risky option, run through four questions before deciding:
- What are the possible outcomes and their probabilities? (Do they add to 100%?)
- What is the payoff (or loss) for each outcome?
- What is the expected value? (Positive, negative, or roughly zero?)
- Is this repeatable and survivable? (If yes, lean on EV. If no, also price in variance and ruin risk.)
The $50,000 campaign passes the first three filters cleanly: net EV of +$30,000. Whether it passes the fourth depends on your cash position and how many similar bets you run each year.
Quick check
Next
Decision trees — chain several decisions and chance events together into a branching diagram, so you can compare multi-stage strategies and find the path with the highest expected value end-to-end.
Practice this in an interview
All questionsExpected value is the probability-weighted average outcome of a random variable; variance measures average squared deviation from that mean. Both are linear/additive in specific ways — knowing these rules prevents algebraic mistakes under interview pressure.
The default 0.5 threshold optimises for balanced accuracy but is rarely the right choice for business objectives. The correct threshold is found by translating the business cost of false positives and false negatives into a cost matrix, then sweeping the threshold on a held-out set to find the point that minimises expected cost or maximises expected profit. Operational constraints — such as review-team capacity — further bound the feasible region.
Optimize precision when a false positive is costly — spam filters, ad targeting, legal evidence — because you'd rather miss some positives than act on wrong ones. Optimize recall when a false negative is costly — cancer screening, fraud detection, safety systems — because missing a true positive can be catastrophic. The business cost of each error type should drive the choice, not the metric itself.
The optimal threshold depends on the business cost of false positives versus false negatives, not on defaulting to 0.5. You choose it by plotting the PR or ROC curve on a held-out set, computing the metric that captures your cost function (e.g., F-beta, revenue, expected cost) at each threshold, and selecting the point that maximises it. Threshold tuning is free and should always precede resampling or model changes.