Expected Value Calculator

Defaults are Win Probability 55 percent, Decimal Odds 1.91, Stake 100. Submit. Expected Value is 5.05 and ROI is 5.05%. Profit if the bet wins is 100 × 1.91 − 100 = 91. EV = 0.55 × 91 − 0.45 × 100 = 50.05 − 45 = 5.05. ROI is EV/stake = 5.05%. Break-even probability at 1.91 is 1/1.91 ≈ 52.356%. Your typed 55% sits 2.644 points of edge above that break-even, which is why the EV is positive.

Lower probability to 52% with the same 1.91 and 100. EV = 0.52 × 91 − 0.48 × 100 = 47.32 − 48 = −0.68, ROI −0.68%. The price did not change; the probability did. Raise odds to 2.10 at 55% and 100: win profit 110, EV = 0.55 × 110 − 0.45 × 100 = 60.5 − 45 = 15.50, ROI 15.5%. Stake 250 at the original 55% / 1.91: EV scales to 12.625, ROI stays 5.05% because both EV and stake quintupled by 2.5.

The probability field is your estimate, not the implied 1/odds. If you type 52.36% at 1.91 you should see EV near 0. If you type the implied probability as if it were your edge, you will manufacture a zero-EV bet and learn nothing. Do not type 55% because it feels lucky; type it because a model or a closing-line study said 55%.

A worked zero: probability 50%, odds 2.00, stake 100 → EV 0, ROI 0, a fair coin at evens. A worked house-edge analogue: probability 48%, odds 2.00, stake 100 → EV −4, ROI −4%, a coin with juice. The sports default (55%, 1.91, 100) is the opposite sign: a small positive after juice if and only if the 55% is true.

Reload 55, 1.91, 100 to recover 5.05 and 5.05%. The tool does not know whether 55% is calibrated. It only multiplies the numbers you accept as inputs. If your model outputs a probability interval 53–57% instead of a point, run both ends: 53% at 1.91 is EV = 100×(0.53×1.91 − 1) = 1.23; 57% is 8.87. The default 5.05 sits in the middle of that band only if 55% is the midpoint you actually believe.

Expected value is the probability-weighted average of profit and loss on a single stake. Huygens’s 1657 treatise On Reasoning in Games of Chance set the fair price of a game at this weighted average. Sports betting uses the same object: if p is the true win chance and decimal odds o are paid, EV = p(o−1)S − (1−p)S. The calculator exists to make that product visible when p, o, and S are known.

Book implied probability 1/o is not p. Edge exists only when p > 1/o (plus any commission). Closing-line value studies treat the market close as a noisy estimator of p; beating the close is evidence, not a proof that tonight’s 55% was true. This tool does not fetch closes. It takes your p as given.

Purpose: separate a good price from a good feeling. A 1.91 favourite you honestly hold at 55% is +EV. A 1.91 favourite you hold at 51% is −EV. Both can win tonight. Only one has a positive mean. Bankroll growth still needs many independent trials; a +5.05 dollar mean on 100 is a 5% ROI, which variance will swamp over a weekend card.

Limits: binary win/lose, no pushes, no each-way, no dead-heat reduction, no Asian-handicap quarter-win splits. Probability is a single number, not a distribution. Stake is constant. Correlation with other bets on the slip is ignored. Huygens assumed known p; you do not have known p unless the process is a physical coin or a certified RNG.

History after Huygens: Pascal, Fermat, and later de Moivre on the binomial; Kelly in 1956 on how much to stake when EV is positive; Thorp on blackjack and then on sports and markets. This calculator stops at the first moment. It will not size the bet (see Kelly) and will not de-vig a book (see margin). It will tell you the mean of the signed profit if your p is the one nature uses.

Expected value is the probability-weighted average of profit and loss on a single stake. Christiaan Huygens’s 1657 treatise on reasoning in games of chance set the fair price of a game at this weighted average. Sports betting uses the same object. If p is the true win chance and decimal odds o are paid on stake S, expected value equals p times (S times o minus S) minus (1 − p) times S. Algebraic rewrite: expected value equals S times (p o − 1). The figure is positive if and only if p exceeds 1/o, the break-even probability implied by the price. Book implied probability 1/o is not p. Edge exists only when your p sits above that threshold after any commission.

The published defaults are a win probability of 55 percent, decimal odds of 1.91, and a stake of 100. Profit if the bet wins is 100 × 1.91 − 100 = 91. Expected value equals 0.55 × 91 − 0.45 × 100 = 50.05 − 45 = 5.05. Return on investment is 5.05 / 100 = 5.05 percent. Break-even probability at 1.91 is 1/1.91 ≈ 52.356 percent. The typed 55 percent sits 2.644 points of edge above that break-even, which is why the mean is positive. Check the compact form: p o − 1 = 0.55 × 1.91 − 1 = 1.0505 − 1 = 0.0505, times 100 equals 5.05, matching both the cash expected value and the return on investment.

Lower probability to 52 percent with the same 1.91 and 100. Then expected value equals 0.52 × 91 − 0.48 × 100 = 47.32 − 48 = −0.68. The price did not change; the probability did. Raise odds to 2.10 at 55 percent and 100: win profit 110, expected value = 0.55 × 110 − 0.45 × 100 = 60.5 − 45 = 15.50, a 15.5 percent return. Stake 250 at the original 55 percent and 1.91: expected value scales to 12.625, return stays 5.05 percent because both the mean and the stake scaled by 2.5. If you type the implied 52.36 percent at 1.91 you should see a mean near zero. Typing the implied probability as if it were your edge manufactures a zero and calls it analysis.

Variance of the signed profit is p (win profit − mean) squared plus (1 − p) (loss − mean) squared, not displayed here. For the defaults it is large relative to 5.05: a single +91 or −100 dwarfs a 5.05 mean. Bankroll growth still needs many independent trials. Closing-line value studies treat the market close as a noisy estimator of p; beating the close is evidence, not a proof that tonight’s 55 percent was true. This worksheet does not fetch closes. It takes your p as given.

Assumptions: one winner, stake lost in full on a loss, odds decimal and net, p the true frequency in the independent-trial sense. If p is a Bayesian credence rather than a frequency, expected value is still the mean of the random variable you defined, but calibration is on you. The worksheet reports expected value and return on investment. Huygens priced a lottery by summing prize times probability. The sports form is that sum with two prizes: plus (o − 1)S and minus S.