Expected Value Calculator: How to Calculate EV Step-by-Step

Expected value (EV) is the average outcome of a decision if you could repeat it many times. To calculate it, you multiply each possible outcome by its probability and add the results. That single sentence is the entire formula — but applying it well is the difference between making good decisions under uncertainty and just hoping for the best.

This guide walks through the EV formula step by step, then works through four real examples — betting, investing, insurance, and a career decision — so you can apply it to any choice you face. If you want the conceptual background first, our expected value explained guide covers what EV means and why it matters before you start calculating.

Expected value is the probability-weighted average of all possible outcomes of a decision. If you flip a fair coin and win £10 for heads or lose £8 for tails, the expected value of one flip is £1 — even though that exact outcome never actually happens on a single flip.

What EV tells you is the average result if you could play the same scenario forever. It does not tell you what will happen on any individual decision. This distinction matters because most real decisions only happen once. A choice with positive EV can still go badly the one time you make it, and a choice with negative EV can still pay off occasionally. Over many decisions, however, EV is what you converge to — and what you should usually optimise for.

The point of calculating EV is to strip the emotional content out of a decision and look at the underlying maths. Once you can see that an insurance policy has a negative EV of -£40 per year, or that a job offer has a positive EV of +£12,000, you have a clear quantitative anchor for the decision rather than gut feeling alone.

The formula for expected value is:

EV = (P₁ × V₁) + (P₂ × V₂) + ... + (Pₙ × Vₙ)

Where:

• P is the probability of each outcome (a number between 0 and 1, where 0 means impossible and 1 means certain).
• V is the value of each outcome (the gain or loss in whatever unit matters — pounds, points, years of life, customer accounts).
• The probabilities of all possible outcomes must sum to 1, because something must happen.

The formula scales to any number of outcomes. For a coin flip there are two terms. For a six-sided die there are six. For a complex investment decision there might be dozens, but the structure is identical: list every possible outcome, assign each one a probability, multiply each value by its probability, and add them together.

If you find yourself listing only the upside outcomes, you are not calculating EV — you are calculating the best case. The hardest part of EV calculation is being honest about the downside cases and their probabilities, especially the ones you would prefer not to think about.

Suppose a friend offers you a bet on a single roll of a fair six-sided die. If it lands on a 6, you win £30. If it lands on anything else, you lose £5. Should you take the bet?

Step 1: List outcomes. There are two: 'rolls a 6' and 'rolls 1-5'.

Step 2: Assign probabilities. P(6) = 1/6 ≈ 0.167. P(1-5) = 5/6 ≈ 0.833. They sum to 1.

Step 3: Quantify values. V(6) = +£30. V(1-5) = -£5.

Step 4: Calculate. EV = (0.167 × £30) + (0.833 × -£5) = £5.00 - £4.17 = +£0.83.

The expected value of one roll is positive — about 83 pence. The bet is favourable in the long run. If you played it 100 times, you would expect to be roughly £83 ahead, on average. On any single roll you will almost certainly either win £30 or lose £5 — never exactly £0.83 — but the EV tells you the underlying maths is on your side.

This worked example shows why EV is the dominant framework in poker, sports betting and any context where the same decision recurs many times. For a deeper application of this idea, our expected value in poker guide walks through pot odds, equity and EV in real game situations.

You are considering investing £10,000 in a small startup. Based on industry base rates and your own assessment, you estimate four possible outcomes over five years.

• 10% chance the startup is acquired and your stake becomes worth £100,000 (gain: +£90,000).
• 20% chance it grows steadily and your stake becomes worth £25,000 (gain: +£15,000).
• 30% chance it survives but doesn't grow — your stake stays roughly flat (gain: £0).
• 40% chance it fails and your stake is worthless (loss: -£10,000).

Probabilities sum to 100%. Calculation:

EV = (0.10 × £90,000) + (0.20 × £15,000) + (0.30 × £0) + (0.40 × -£10,000)

EV = £9,000 + £3,000 + £0 - £4,000 = +£8,000

The expected value over five years is +£8,000. That sounds attractive — but notice how the calculation depends entirely on those probability estimates. If the failure rate is actually 60% rather than 40%, the EV drops to -£2,000 and the decision flips. Sensitivity analysis (recalculating with different probability assumptions) is essential for real investment EV — see our risk vs uncertainty guide for why this matters when probabilities themselves are uncertain.

An extended-warranty company offers a £80 three-year warranty on a £400 laptop. Based on published failure rates, you estimate there is roughly a 12% chance of a covered failure in three years, and the average claim value would be £200.

• 12% chance of a covered failure: claim value £200, minus the £80 premium = +£120 net.
• 88% chance of no claim: -£80 (the premium you paid).

EV = (0.12 × £120) + (0.88 × -£80) = £14.40 - £70.40 = -£56

The expected value is negative £56. On average, you lose money buying this warranty — which is why insurance companies sell warranties profitably. This does not automatically mean you should refuse the warranty. EV ignores variance, and there is a meaningful psychological case for paying a small premium to eliminate the small chance of a £200 surprise. But the calculation puts a price on that comfort: roughly £56 over three years, or £19 per year.

The same logic applies to most insurance products — health, travel, gadget, pet. The product is sold at a negative EV by design (otherwise the insurer loses money). The question is not 'is the EV positive?' but 'is the EV negative enough that I'd rather take the risk myself, or close enough to zero that the peace of mind is worth it?'

You have two job offers. Job A pays £55,000 and is at a stable, established company. Job B pays £45,000 base plus equity in an early-stage startup. You want to compare them on three-year EV.

Job A (stable):

• 95% chance you stay employed and earn the salary as expected: 3 × £55,000 = £165,000.
• 5% chance of redundancy with 6 months severance: ~£82,500 over the period.

EV(A) = (0.95 × £165,000) + (0.05 × £82,500) = £156,750 + £4,125 = £160,875

Job B (startup with equity):

• 10% chance of acquisition where equity is worth £200,000: salary £135,000 + equity £200,000 = £335,000.
• 30% chance of growth where equity is worth £30,000: £135,000 + £30,000 = £165,000.
• 40% chance equity is worthless but you keep the salary: £135,000.
• 20% chance of startup failure mid-period (lose 18 months of expected salary): £67,500.

EV(B) = (0.10 × £335,000) + (0.30 × £165,000) + (0.40 × £135,000) + (0.20 × £67,500) = £33,500 + £49,500 + £54,000 + £13,500 = £150,500

On pure EV, Job A wins by about £10,000 over three years. But the EV calculation hides important non-financial factors — career capital from startup experience, learning rate, network value, optionality. EV is the financial anchor, not the whole answer. For more on integrating non-financial factors with EV, see our probabilistic framework for career decisions.

1. Missing outcomes. If your probabilities don't sum to 1, you've forgotten a case. The most commonly missed outcome is the boring middle case — 'nothing much happens' — because it's not interesting to imagine. This systematically biases EV upward.

2. Anchoring probabilities to the first number you saw. If you read that 'most startups fail', you might assign a 90% failure rate without checking. Real base rates vary enormously by sector, stage and team experience — see our anchoring bias guide for why first numbers stick.

3. Conflating one decision with many. EV is the average of many trials. If you only get one shot at the decision, the variance matters as much as the mean. A +£8,000 EV with a 40% chance of losing £10,000 is not the same as a guaranteed +£8,000 — the latter is much better even though the EV is identical.

4. Ignoring ergodicity. Some negative outcomes can take you out of the game permanently (a bet that requires you to risk all your savings). In those cases, even a positive EV can be the wrong choice — a topic our ergodicity explained guide goes into depth on.

5. Treating EV as a forecast. EV is a long-run average, not a prediction of what will happen this time. The single biggest misuse of EV is presenting it as 'what will probably happen' rather than 'what would happen on average if this decision were repeated many times'.

Rule 1: Use EV for decisions you'll face many times. Subscription cancellations, recurring purchases, betting decisions, repeated investments — anywhere the same decision pattern recurs, EV is the highest-leverage framework. The more repetitions, the more EV converges to your actual long-run result.

Rule 2: For one-off decisions, calculate EV but also think about variance. A positive EV with a 50% chance of catastrophic loss is rarely the right call for a one-shot decision. Combine EV with the Kelly criterion for sizing decisions where you have an edge but limited bankroll.

Rule 3: When you can't calculate EV precisely, calculate it roughly. A back-of-envelope EV with rough probabilities and round numbers is often enough to flip a decision from 'feels wrong' to 'is clearly wrong' — or vice versa. Don't let the search for precise probabilities prevent you from doing the calculation at all. A rough EV beats no EV every time.