A bet can win and still have been a bad bet. Expected value calculation is how you separate the two. The lesson starts with the formula for a two-way wager, then walks a +120 price through it using two different probability estimates, showing that a four-point change in your estimate flips a gain into a loss. You also convert a probability into fair odds, find the break-even point, and handle pushes with a three-outcome payoff table. The big takeaway for a bettor: a generous-looking payout is not value on its own. Value depends on whether your probability holds up. Write down price, probability and edge separately every time, and your notes get honest fast.
Learning objectives
- Explain this technique and its assumptions.
- Reproduce the worked numerical example and interpret its limitations.
- Apply a documented research workflow and decide when evidence is insufficient.
Guided lesson
Expected value (EV) is a weighted average of all possible net outcomes. It is not a forecast that one selection will win. For a two-way wager at decimal odds d, with win probability p and one unit staked, EV = p × (d − 1) − (1 − p) = p × d − 1. A true probability estimate is essential: a bookmaker's quoted probability is not automatically the true probability.
Worked example
Imagine a fictional selection at +120, decimal 2.20. A one-unit stake would profit 1.20 units if it wins and lose one unit if it loses. If independently estimated p=.48, EV=.48×1.20−.52=+.056 units per unit staked, or +5.6% modeled ROI. Break-even probability is 1/2.20=45.45%. At p=.44 instead, EV=.44×2.20−1=−3.2%. The estimate changing by four percentage points flips the sign of the result.
Fair decimal odds, ignoring fees and pushes, equal 1/p. At 48%, fair decimal odds are 2.0833, or approximately +108 American. Do not call +120 'value' simply because the payout seems generous; the value exists only to the extent that the 48% estimate is defensible. Distinguish price, probability, and edge in every research note.
A push requires a three-outcome payoff table instead of the simple binary formula. If a selection can win, lose or push, EV = P(win)×net_win + P(loss)×net_loss + P(push)×0. Some products have partial payouts or fees that need their own outcomes. When prices and probabilities lack matching settlement definitions, the calculation is not valid.
Practice
A fictional contract at decimal 1.90 has estimated win probability 54%. Gross EV is .54×1.90−1=+2.6% on stake. At 51% probability EV is −3.1%. Check: Which estimate should be published? Only the one justified by the source model, with an uncertainty range.
Applied tutorial: In Sharp App's +EV research workflow, record the selection, sportsbook price, timestamp, fair-price source, market rules, and probability uncertainty; confirm the market is actually available. Reproduce the formula in a calculator. Treat small modeled positives as fragile until fees, model disagreement, and quote age are checked.
Capstone reflection
Explain which specific conditions would make the lesson’s technique invalid in a real market, and how you would audit those conditions before comparing prices.
Recap
- Expected value weights every net outcome by its probability: EV equals p times d minus 1, and fair decimal odds equal 1 divided by p.
- At +120 a 48% estimate gives +5.6% modeled EV and a 44% estimate gives -3.2%, so a four-point change flips the sign.
- Pushes, partial payouts and fees need their own outcomes, and a generous-looking payout is not value without a defensible probability.
Frequently asked questions
What is the expected value formula in betting?
For decimal odds d and win probability p, one unit staked has EV equal to p times d, minus 1. It weights every net outcome by its probability. It is not a forecast that one bet wins, and it is only as good as your probability estimate.
How do I convert a probability to fair odds?
Fair decimal odds equal 1 divided by your win probability, ignoring fees and pushes. At 48%, fair decimal odds are 2.0833, about +108 American. A posted +120 is only value if the 48% is defensible.
How do I find the break-even probability?
Divide 1 by the decimal odds. At +120, which is decimal 2.20, break-even is 1 / 2.20 = 45.45%. Your estimate has to beat that for EV to be positive, and a four-point change in your estimate can flip the sign.
How does a push change the EV calculation?
A push needs a three-outcome table: EV equals P(win) times the net win, plus P(loss) times the net loss, plus P(push) times zero. Partial payouts and fees need their own outcomes. If prices and probabilities use different settlement rules, the math is not valid.
Related lessons
- Player Prop Projection Model vs Market Probability
- Advanced +EV Filters and Research Workflow
- Pick'em Expected Value: Break-Even Math Explained
- DFS Pick'em strategy — Underdog Fantasy + similar
Sources and further reading
- Expected value (overview)
- About Sharp App: who builds these lessons and how the tools work.
Try it in the +EV tracker. Examples in this lesson are fictional, built to show the method. Work the numbers by hand first, then repeat the same steps on a real board. You can also read how the EV Tracker works on the Sharp App site.
Questions? Ask ARBAI about this lesson · Ask our Sharps in Discord →
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