Expected value

Expected value (EV) is the long-run average outcome of a bet, expressed in dollars or as a percentage of stake. Positive EV means a bet has positive expected long-run profit. Negative EV means it has expected long-run loss. EV is the only rigorous answer to 'is this bet worth placing.'

The formula

Expected value is the probability-weighted average of the outcomes of a bet.

EV = (P_win × profit_if_win) + (P_lose × loss_if_lose)

Where:
  P_win  = your estimated probability the bet wins
  P_lose = 1 - P_win
  profit_if_win  = stake × (decimal odds - 1)
  loss_if_lose   = -stake

An example. You bet $100 at +110 (decimal 2.10) on a side you estimate to be a 55% favorite.

STAKE              $100
DECIMAL ODDS       2.10
YOUR PROB          0.55

EV = (0.55 × $110) + (0.45 × -$100)
   = $60.50 - $45.00
   = +$15.50 expected profit per $100 staked

EV%  = $15.50 / $100 = +15.5%

On average, that bet pays you $15.50 for every $100 staked. In any single instance you win $110 or lose $100. Over thousands of repetitions of bets like this, the average converges on +$15.50.

Where the probabilities come from

The EV formula has two terms. The odds are observable on the sportsbook. The probability is your estimate of how likely the side is to win. That estimate is the entire game.

• If you have a working model, the probability comes from the model.
• If you do not have a model, the probability comes from a process: comparing the line to a sharp book (Pinnacle is the standard reference), to consensus across books, or to your own informed view.
• If your probability estimate is just 'I feel good about this team,' your EV calculation is decoration on a guess.

The honest version is: you cannot have positive expected value without a probability estimate that is more accurate than the market's, in the long run, on the bets you actually place.

Implied probability vs. fair probability

The implied probability of a bet is what the odds say the probability is, ignoring the book's hold. The fair probability (sometimes called no-vig probability or true probability) removes the hold and treats the two sides as summing to exactly 100%. The fair probability is a better estimate of what the market actually thinks.

MARKET
  Side A:  -120  →  implied 54.5%
  Side B:  +100  →  implied 50.0%
  total              104.5%   (4.5% hold)

REMOVE THE HOLD (normalize to 100%)
  fair side A = 54.5 / 104.5 = 52.2%
  fair side B = 50.0 / 104.5 = 47.8%

INTERPRETATION
  The market thinks side A is a 52.2% favorite
  after stripping out the book's margin.

Sharp books (Pinnacle, Circa) run hold close to 2%, so their fair probability is close to their implied probability. They are the cleanest reference for the market's actual estimate. Most US retail books run 4 to 5% hold on a balanced market. The implied probability you read off the screen is inflated.

EV and hit rate are not the same thing

A bettor with a 60% hit rate at -200 odds is losing money. A bettor with a 40% hit rate at +200 odds is making money. Hit rate without odds is uninterpretable.

BETTOR A: 60% hit rate at -200 (decimal 1.50)
  EV per $100 stake
    = (0.60 × $50) + (0.40 × -$100)
    = $30 - $40
    = -$10  (negative EV, slowly bleeding)

BETTOR B: 40% hit rate at +200 (decimal 3.00)
  EV per $100 stake
    = (0.40 × $200) + (0.60 × -$100)
    = $80 - $60
    = +$20  (positive EV, slowly compounding)

The breakeven hit rate at any price is the implied probability of that price. At -110, the breakeven is 52.4%. At +200, the breakeven is 33.3%. Anything above breakeven on consistent prices is positive EV.

From EV to expected value over a season

Bettors with consistent positive EV around 2 to 5% per bet (which is realistic for a sophisticated operator at scale) compound into meaningful returns over a season. Bettors with marginal positive EV around 1% are functionally breakeven once you account for variance.

SCENARIO: 1,000 bets at $100 each
  +1% EV    →   +$1,000 expected profit
  +3% EV    →   +$3,000 expected profit
  +5% EV    →   +$5,000 expected profit
  -2% EV    →   -$2,000 expected loss

  ROI on handle, not on bankroll.
  Bankroll growth depends on sizing.

These numbers compound. A bettor working at 3% EV with disciplined sizing can outperform most equity funds. A bettor working at -2% EV (which describes most retail bettors) can lose half their bankroll in a year without doing anything dramatic.

What to read next

EV is the framework. Closing line value is the metric most operators track to verify they are actually betting positive EV in practice. Kelly criterion is how to size a positive-EV bet to maximize bankroll growth without overbetting.