Learning goals and lesson guide
**Estimated reading time:** 9 min read.
**Level:** Intermediate. **Study time:** Allow 35–45 minutes with a calculator. **Prerequisites:** Decimal returns, implied probability, settlement rules and paper records. All prices and probability estimates below are fictional teaching inputs.
A team may be more likely to win than lose while its offered price still produces a negative expected result under your model. Expected value connects the size of each possible outcome with its probability. It does not identify what will happen in the next match, and a positive calculation is only as credible as the assumptions behind it.
By the end of this lesson, you should be able to calculate expected net results for simple win/loss and push examples, identify a break-even probability and show how uncertainty can reverse a conclusion. You will also learn to separate price-derived probabilities from independent estimates and write a conditional conclusion instead of claiming guaranteed profit.
## 1. Define the possible net outcomes first
Use a one-unit hypothetical cash stake and decimal odds of 2.00. A win returns two units, including the stake, so net profit is one unit. A loss returns zero, so the net result is minus one unit. These are the quantities to weight by probability.
At odds of 1.80, the net win is 0.80 units and the net loss remains minus one. At 3.00, the net win is two units. Confusing total return with net profit produces an overstated expected value because it counts the original stake twice.
For a model with outcomes indexed by i, expected net result is the sum of each outcome's net value multiplied by its probability. The probabilities must cover all relevant outcomes and add to one. If pushes or partial settlements are possible, a two-outcome formula may be incomplete.
## 2. Derive the simple formula
Let p be the assumed win probability and d the decimal odds. For one unit staked with only win or loss, expected net result is p × (d − 1) + (1 − p) × (−1). Simplifying gives p × d − 1.
At p = 0.55 and d = 2.00, the result is 0.55 × 2 − 1 = 0.10 units per unit staked. With a hypothetical PHP 100 stake, that is PHP 10 expected net under the model. The actual single-event result remains plus PHP 100 or minus PHP 100, not PHP 10.
This distinction is central. Expected value is an average implied by a probability model across repeated comparable opportunities, not a smaller guaranteed payment from one event. The model may also be wrong, and actual opportunities may not remain comparable over time.
## 3. Find the break-even probability
Set p × d − 1 equal to zero. The resulting threshold is p = 1 ÷ d for the simple no-push, no-fee cash-stake example. At 1.80, it is approximately 55.56%. At 2.50, it is 40%.
This threshold tells you what probability would make the model's expected net zero at that price. It does not prove that the outcome has that probability. A sportsbook's reciprocal price can contain margin and need not match an independent estimate of the true chance.
| Decimal price | Simple break-even probability | | --- | --- | | 1.50 | 66.67% | | 1.80 | 55.56% | | 2.00 | 50.00% | | 2.50 | 40.00% |
If the market has commission, deductions, pushes or special promotional return rules, derive the threshold from its actual net outcomes. Do not reuse the simplest formula without checking its assumptions.
## 4. Test sensitivity to the probability estimate
Suppose a fictional price is 1.90. Compare three assumptions:
| Assumed win probability | Expected net per unit | | --- | --- | | 50% | −0.050 | | 53% | +0.007 | | 56% | +0.064 |
The 53% case produces only 0.7% of stake in modelled expected net. A small probability error can erase that result. Rounding an uncertain estimate to a confident-looking percentage does not make it more reliable.
If a plausible range is 50%–56%, the expected-result range crosses zero. A careful note says the conclusion is sensitive to the estimate. It does not select the top of the range and report that figure as established. State how the range was chosen; an arbitrary range is only an illustration.
## 5. Include pushes explicitly
Assume a fictional whole-number market at 2.00 with 45% win, 10% push and 45% loss. Net outcomes are +1, 0 and −1 units. Expected net is 0.45 × 1 + 0.10 × 0 − 0.45 × 1 = zero.
The unconditional win probability is only 45%, yet the example breaks even because pushes return the stake. Among decisive outcomes, the win probability is 45 ÷ 90 = 50%. This demonstrates why inserting the unconditional win probability into the no-push formula would give the wrong answer.
For general win probability w, push probability r and loss probability l, with w + r + l = 1, expected net is w × (d − 1) − l. At break even, w × (d − 1) = l. Keep the complete outcome structure visible.
## 6. Handle partial outcomes as a distribution
Quarter lines can produce full wins, half wins, half losses and full losses. Assign each its correct net result, then weight by the modelled probability. Do not force every outcome into a binary win/loss count.
At 1.90 on one unit, the net values are +0.90 for a full win, +0.45 for a half win, −0.50 for a half loss and −1 for a full loss. A push, where possible, contributes zero. The probabilities depend on the underlying score distribution and the specific line.
For an invented distribution of 40% full win, 20% half win and 40% full loss, expected net is 0.40 × 0.90 + 0.20 × 0.45 − 0.40 = +0.05 units. This arithmetic does not establish that the invented probabilities are realistic. It only shows how to use them consistently.
## 7. Do not confuse overround with your expected loss
Adding reciprocal prices across all outcomes describes the market's overround under a familiar convention. It is useful for understanding pricing, but it does not independently reveal the true probabilities of every outcome or the expected result of your particular selection.
Normalising the reciprocals to sum to one creates one possible price-based baseline. It is not proof that the margin was distributed proportionally. Different assumptions can produce different adjusted estimates, especially across outcomes with very different prices.
Use an adjusted market baseline as a labelled model. Compare it with your own estimate only when both refer to the same event, period and settlement terms. A disagreement is a research question, not automatic evidence that your estimate is superior.
## 8. Evaluate how the estimate was produced
An unsupported 60% guess remains unsupported after being placed in a correct equation. Ask whether the estimate uses relevant historical data, verified current information and a method tested on later unseen events. Check whether the method was chosen only after inspecting favourable results.
Probability uncertainty can come from small samples, changing rosters, model assumptions, missing features or data errors. These are different from the randomness of the match itself. A model can be confidently wrong if it ignores an important change.
Write down what would invalidate the estimate. For example, a forecast assuming a player's usual role should be reviewed if participation is unconfirmed. Do not retain the same percentage merely because the arithmetic has already been completed.
## 9. A complete paper analysis
Use a fictional price of 2.20 and three assumed probabilities: 42%, 46% and 50%. Expected net per unit is −0.076, +0.012 and +0.10 respectively. The simple break-even probability is about 45.45%.
A responsible conclusion is: “At the central illustrative estimate of 46%, the modelled expectation is slightly positive, but the range includes a negative result. The probability inputs require validation, so this is a sensitivity exercise rather than a recommendation.”
Add a second version with a stated two-percent fee on net winnings only. The net win becomes 1.20 × 0.98 = 1.176 units, while a loss remains minus one. At p = 0.46, expected net is 0.46 × 1.176 − 0.54 = 0.00096 units. This shows how a small cost can nearly erase a small calculated difference. Actual fee structures must be checked separately.
## 10. Practice questions and explained answers
1. What is expected net per unit at p = 0.60 and d = 1.60? 2. What is the simple break-even probability at 2.40? 3. Does +0.10 expected net mean the next event pays +0.10? 4. Why must a push be included separately? 5. At 2.00, a model assigns 40% win, 20% push and 40% loss. What is expected net? 6. Does a positive calculation validate the probability estimate? 7. What does a sensitivity range crossing zero imply?
### Answer key
1. 0.60 × 1.60 − 1 = −0.04 units. A likely winner can still have a negative modelled expectation at the price. 2. 1 ÷ 2.40 ≈ 41.67%, under the simple assumptions. 3. No. It is a probability-weighted average, not a single-event payment. 4. A push has zero net result and reduces the probability allocated to decisive outcomes. 5. Zero: 0.40 × 1 − 0.40 × 1. 6. No. Correct arithmetic cannot establish that the model inputs are accurate. 7. The direction of the conclusion depends on uncertain assumptions; report that sensitivity rather than selecting only the favourable endpoint.
## 11. Completion check
Submit a one-page paper analysis containing the exact market, net outcomes, probability assumptions, calculation, sensitivity table and limitations. Another learner should be able to reproduce every number. A finding of uncertain or negative expectation is a valid analytical result.
Review [probability and margins](https://betting.crazywingo.ph/article/implied-probability-bookmaker-margins-beginners), [returns](https://betting.crazywingo.ph/article/read-betting-odds-calculate-payouts), [market types](https://betting.crazywingo.ph/article/moneyline-spread-total-parlay-beginners), [market identification](https://betting.crazywingo.ph/article/sports-betting-basics-for-beginners) and [budget boundaries](https://betting.crazywingo.ph/article/betting-budget-limits-avoid-chasing-losses) as needed. Next comes the variation that can separate actual results from a model's expectation.