Expected value under uncertainty
Expected value is one line of arithmetic, but it is only as good as the probability you feed it. Learn to compute EV, break-even and fair odds, then test whether an apparent edge survives a realistic range of estimates.
Learning objectives
- Compute EV, break-even probability, probability edge and fair odds from a price and a probability.
- Show that EV moves by o percentage points for every percentage point of probability error.
- Build a sensitivity band and classify a price as positive across the range, negative across the range or dependent on the estimate.
- Explain why a Kelly stake computed on a point estimate overstates the stake when the estimate is uncertain.
Expected value in one line
EV = p × o − 1Break-even p* = 1/o Edge = p − 1/o Fair odds = 1/pEV is positive exactly when your probability is above the break-even bar. It is the average result per unit across many independent repetitions if your probability is right. It is a property of your belief about the bet, not a property of the bet itself, and it says nothing about what will happen next time.
EV = o × (p − 1/o)Notice that break-even uses the raw price 1/o, not a margin-free probability. The margin is already inside the price you are offered, so your own probability has to clear the full bar. Margin-free probabilities (lesson 3) answer a different question: what the market as a whole seems to believe. A large gap between your estimate and that belief is a warning sign before it is an opportunity.
EV is linear in p — and so are your errors
Change in EV = o × change in pAt odds of 2.30, a one-point error in p moves EV by 2.3 points. At odds of 10.0 it moves EV by 10 points. Longer odds amplify probability errors, which is one reason long-shot “edges” are so fragile.
| Probability | EV per unit |
|---|---|
| 40% | −8.0% |
| 42% | −3.4% |
| 44% | +1.2% |
| 46% | +5.8% |
| 48% | +10.4% |
| 50% | +15.0% |
| 52% | +19.6% |
Because EV is a straight line in p, its lowest and highest values over any probability interval sit at the two ends of that interval. That is why the Fair Odds and EV lab only needs to evaluate the low end, the estimate and the high end.
A forecast is an interval, not a point
A figure like 48% comes from a model with estimated parameters, a limited sample and incomplete team news. State it as a range you would honestly defend, then classify the price. Positive across the range: even the low end clears break-even. Negative across the range: even the high end falls short. Depends on the estimate: the interval straddles break-even.
Where does the width come from? At least three places. Parameter uncertainty: team strengths estimated from a few dozen matches are noisy. Model choice: two reasonable models, or two de-vig methods, can disagree by half a point or more. Information timing: a lineup announced after you formed the estimate can move the true chance by more than your whole edge. Account for all three honestly; it is easy to draw a first interval that is too narrow.
Move the interval and watch the verdict change.
Lab 1 · Fair odds and expected value
Synthetic · deterministicNo clear advantage. Your range straddles break-even. Whether the bet has value depends on which end of your estimate is right, so there is no clear advantage. A careful analyst treats this as a pass unless better evidence narrows the range.
Show data table
| True probability (%) | Expected ROI |
|---|---|
| 32% | -26% |
| 33% | -24% |
| 34% | -22% |
| 35% | -20% |
| 36% | -17% |
| 37% | -15% |
| 38% | -13% |
| 39% | -10% |
| 40% | -8% |
| 41% | -6% |
| 42% | -3% |
| 43% | -1% |
| 44% | 1% |
| 45% | 3% |
| 46% | 6% |
| 47% | 8% |
| 48% | 10% |
| 49% | 13% |
| 50% | 15% |
| 51% | 17% |
| 52% | 20% |
| 53% | 22% |
| 54% | 24% |
| 55% | 26% |
| 56% | 29% |
| 57% | 31% |
| 58% | 33% |
| 59% | 36% |
| 60% | 38% |
| 61% | 40% |
| 62% | 43% |
| 63% | 45% |
| 64% | 47% |
Educational calculator. Uses only the numbers you enter or seeded synthetic data — no live prices, no staking, no real money.
From EV to stake size, on a simulated bankroll
Kelly f* = (p × o − 1) / (o − 1) = EV / (o − 1)Kelly's result assumes the probability is known Kelly (1956). At 2.30 the Kelly share of a simulated bankroll is 8.0% at p = 48%, 0.92% at p = 44% and zero at p = 42%. A six-point probability range moves the stake from nothing to 8%, so stake size is far more fragile than the sign of EV. Value 7 itself uses fixed free-Credit stakes; these percentages are for simulation only, and Kelly and fractional Kelly covers the details.
One simple discipline for simulations is to size from the low end of your interval rather than from the point estimate: at 44% the Kelly share is 0.92% instead of 8.0%, and at 42% it is zero. This is a teaching heuristic that respects uncertainty, not the estimator Baker & McHale derive, and the robust decisions lesson compares it with more formal approaches.
Worked example
Worked example · Ashbridge at a hypothetical 2.30
A hypothetical quote of 2.30 on Ashbridge to beat Millford. Your model says 48%, and you believe the true chance plausibly lies between 42% and 54%.
- Break-even
1 / 2.3043.48% - Probability edge
48% − 43.48%+4.52 percentage points - EV at the estimate
0.48 × 2.30 − 1 = 1.104 − 1+10.4% - Fair odds
1 / 0.482.083 - Kelly on the point estimate
0.104 / (2.30 − 1) = 0.104 / 1.308.0% of a simulated bankroll - If the true chance is 42%
0.42 × 2.30 − 1 = 0.966 − 1−3.4% - Sensitivity band 42%–54%
EV at 42% = −3.4%; at 54% = +24.2%Depends on the estimate: no clear advantage - Narrower band 44%–52%
EV at 44% = +1.2%; at 52% = +19.6%; Kelly at 44% = 0.92%Positive across the range, but only just at the low end
A headline +10.4% EV hides a range from −3.4% to +24.2%. Whether this price is worth taking depends entirely on how confident you can honestly be in 48%. With a ±6-point interval the right answer is “no clear advantage”; with ±4 points the edge survives, but the low-end EV of +1.2% and Kelly share of 0.92% show how thin it is.
Practice · Stress-test an edge
Open the Fair Odds and Expected Value lab with odds 2.30, estimate 48% and range 42%–54%. Note the verdict. Narrow the range to 44%–52%. Finally set the estimate to 45% with range 40%–50%.
Lab 1 · Fair odds and expected value
Synthetic · deterministicNo clear advantage. Your range straddles break-even. Whether the bet has value depends on which end of your estimate is right, so there is no clear advantage. A careful analyst treats this as a pass unless better evidence narrows the range.
Show data table
| True probability (%) | Expected ROI |
|---|---|
| 32% | -26% |
| 33% | -24% |
| 34% | -22% |
| 35% | -20% |
| 36% | -17% |
| 37% | -15% |
| 38% | -13% |
| 39% | -10% |
| 40% | -8% |
| 41% | -6% |
| 42% | -3% |
| 43% | -1% |
| 44% | 1% |
| 45% | 3% |
| 46% | 6% |
| 47% | 8% |
| 48% | 10% |
| 49% | 13% |
| 50% | 15% |
| 51% | 17% |
| 52% | 20% |
| 53% | 22% |
| 54% | 24% |
| 55% | 26% |
| 56% | 29% |
| 57% | 31% |
| 58% | 33% |
| 59% | 36% |
| 60% | 38% |
| 61% | 40% |
| 62% | 43% |
| 63% | 45% |
| 64% | 47% |
Educational calculator. Uses only the numbers you enter or seeded synthetic data — no live prices, no staking, no real money.
Show what you should find
Break-even stays at 43.48% throughout. At 42%–54% EV runs from −3.4% to +24.2%: depends on the estimate. At 44%–52% it runs from +1.2% to +19.6%: positive across the range. With estimate 45% and range 40%–50%, EV is +3.5% at the estimate but −8.0% to +15.0% across the range: depends on the estimate. The verdict is driven by the width of your uncertainty as much as by your point estimate.
Common mistakes
- ✗ Reading +10.4% EV as a forecast of results.EV is a long-run average under your probability. A single pick at 2.30 either returns +1.30 or −1 per unit, and short runs vary widely.
- ✗ Ignoring the uncertainty on p.State a range and evaluate EV at both ends. If the range straddles break-even, record “no clear advantage”.
- ✗ Applying full Kelly to a point estimate.Kelly assumes p is exact. With estimation error, shrink the stake or skip the pick; Baker & McHale give the reasoning.
- ✗ Confusing probability edge with EV.A 4.52-point probability edge at 2.30 is a 10.4% EV, because EV = o × (p − 1/o). The same point edge means more EV at longer odds, and so does the same error.
When it fails
Supporting research
- Kelly (1956): A New Interpretation of Information RateBell System Technical Journal · Peer-reviewed theoryMaximising expected log wealth gives the fastest long-run growth when probabilities are known exactly; the Kelly fraction follows from this.
- Baker & McHale (2013): Optimal Betting Under Parameter Uncertainty: Improving the Kelly CriterionDecision Analysis · Peer-reviewed theoryWhen probabilities are estimated with error, Kelly stakes should be shrunk; shrunken strategies did better out of sample in their simulations and tennis illustration.
- Sauer (1998): The Economics of Wagering MarketsJournal of Economic Literature · Literature reviewPrices are, to a first approximation, efficient forecasts, so an estimate far from the margin-free price should be treated as a likely model error before it is treated as an edge.
Check your understanding
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