LineExpected value in one paragraph
The expected value of a bet is what it returns on average if you could place it a million times. A $100 bet at +100 that truly wins 52% of the time returns, on average: (0.52 × $100) − (0.48 × $100) = +$4 per bet. That's a +EV bet — each individual result is a coin flip, but the price is wrong in your favor, and over volume the wrongness becomes money. A -EV bet is the same machine running in reverse, which is what nearly every casually placed bet is, because the vig starts everyone below zero.
The formula: EV = (win probability × profit if won) − (loss probability × stake). The hard part was never the arithmetic. It's the win probability — where does that 52% come from?
The no-vig line: your first honest probability
The cleanest source is the market itself, with the juice stripped out. Take a sharp book's two-sided price — say -115 / -105. Convert both to implied probability (53.5% and 51.2%), which sum to 104.7%; the extra 4.7% is the vig. Divide each side by the total: the no-vig probabilities are 51.1% and 48.9%. That's the sharpest available estimate of the true odds. Now the +EV hunt has a concrete form: find any book, promo, boost, or slow-moving line offering a price better than the no-vig number. If the fair probability is 51.1% and somewhere is paying +105 on that side (implied 48.8%), you've found measurable positive expectation without needing an opinion about the game at all.
This is why most systematic +EV betting looks less like sports analysis and more like retail arbitrage: scanning prices, comparing to a fair benchmark, and buying mispriced inventory. Boosts, mispriced props, and slow lines at soft books are the shelf. The handicapping route to +EV — building your own probabilities that beat the market's — is real but much harder, and the only way to know you're on it is closing line value.
The three ways people fool themselves
Confusing +EV with winning tonight. A +4% edge loses 48 times out of 100. The edge lives in the price, not the outcome, and it needs hundreds of bets to reliably show up in the bankroll. People quit +EV strategies during ordinary variance constantly — the math was never the hard part; the emotional runway is.
Grading their own homework. If your "true probability" comes from your own gut, EV calculations become a machine for laundering hope into decimals. Anchor to no-vig market prices, or to a model with a tracked record — never to how confident you feel.
Ignoring the limits problem. Books limit winners, and they find them fast (mostly via CLV). A +EV strategy's real-world ceiling is how much volume you can get down before the accounts shrink. That's not a reason to skip it — it's a reason to treat it as a grind with a shelf life, not a retirement plan.