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Illustrative calculations only. Prices and probabilities entered here are not verified event forecasts or offers.
The definition of expected value
Expected value describes an arithmetic average over hypothetical repetitions when the probability model is correct. For a two-outcome fixed-price decision with win probability p, decimal odds d, and stake s, net expected value is s multiplied by [p(d−1)−(1−p)]. The calculation treats a win as net profit s(d−1) and a loss as −s.
Suppose a person enters a 45 percent estimated chance, decimal odds 2.40 and a 10-unit stake. A win profits 14 units; a loss loses 10 units. Expected net result under those assumptions is 0.45×14 − 0.55×10 = 0.80 units. The figure is conditional on the input probability, not evidence that an actual market is profitable.
Break-even probabilities and margins
The break-even probability for a single fixed-price bet is 1/d. At odds 2.40 that is roughly 41.67 percent. The difference between an entered probability and the break-even threshold drives the modelled edge. A calculation may be positive, negative or zero; the calculator displays all three without hiding unfavourable scenarios.
Bookmaker odds can include overround, and quoted odds can change. A single price cannot establish a fair value estimate without an independent probability model. The calculator does not estimate probabilities from public rumours, past streaks, promotion graphics, or untested forecasts. It simply uses the probability entered by the user.
Sensitivity analysis and forecast uncertainty
A small adjustment in estimated probability can reverse an expected-value result. In the example at decimal 2.40, changing the estimated win probability from 45 percent to 40 percent changes expected net value per 10-unit stake from +0.80 to −0.40 units. Presenting only the positive case would conceal the importance of model uncertainty.
The interface therefore also displays a sensitivity range using probability estimates five percentage points below and above the chosen value, constrained to zero and 100 percent. This is a scenario comparison, not a scientific confidence interval. Independent assessment would require transparent data collection, calibration, validation and an explanation of model limitations.
Return on stake and variability
The expected-value percentage is expected net value divided by the amount staked. It is not a guaranteed return on a single event or proof of an achievable long-run advantage. Even with a genuinely positive expectation, a run of losses can occur, and an uncertain estimate may be badly biased.
A calculator should not infer an exact probability of eventual profit from the average alone. Risk depends on probability, payoffs, number of decisions, dependence among events and stake policy. This tool intentionally avoids showing invented success rates or deterministic claims about future outcomes.
Conditions, scope and educational use
Inputs must be a positive stake, decimal odds greater than one and an estimated probability between zero and one hundred. Missing or non-finite entries return a visible input message. The calculation assumes exactly two mutually exclusive outcomes—full win or full loss—and excludes partial settlement, cash out, void outcomes, taxes, fees and bonus credits.
For a refund, each-way product, correlated multi-leg bet or unusual settlement rule, a richer event tree would be needed. As a standalone educational tool, this calculator does not constitute an invitation to wager, an investment product, or a claim that a positive estimated EV can actually be obtained.
An alternative expected-value scenario
Take a hypothetical event quoted at decimal odds 1.80 with a 60 percent estimated success probability and a 20-unit stake. A win produces 16 units net profit, while a loss costs 20 units. The assumed net expectation is 0.60 × 16 − 0.40 × 20 = 1.60 units. If a better-calibrated estimate later places success at only 50 percent, the assumed expectation becomes 0.50 × 16 − 0.50 × 20 = −2 units.
The tool does not infer that 60 percent is a good estimate, nor does it track actual bookmaker movements. The scenario highlights why a precise-looking result can be dominated by uncertainty in the least reliable input. If outcomes could draw, partially settle, or have asymmetric fees, the simple win-loss equation would not be sufficient. The correct next step is to model each possible net payoff and its probability rather than forcing them into two outcomes.
Why a positive calculation is not a betting recommendation
A forecast should be assessed on out-of-sample performance rather than chosen because it produces a positive answer. If probabilities are estimated by inspecting the same outcomes that were used to select a method, the apparent accuracy can be exaggerated. A robust review would specify a training period, independent evaluation period, consistent market definitions and a record of all selections rather than a collection of favourable examples. The output shown above is a conditional arithmetic result and has no such model-quality certification.
Sources and further reading
The worked calculations are original illustrations. Official operator rules and regulatory information help distinguish mathematical outcomes from actual product terms.
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