Expected Value Betting Calculator
Calculate the expected value of a bet from your estimated probability and offered odds. Enter values for instant results with step-by-step formulas.
Calculator
Adjust values & calculateSimulation: 100 Bets
Formula
Expected Value equals the probability of winning multiplied by the net profit if you win, minus the probability of losing multiplied by the amount you lose. A positive EV means the bet is profitable long-term. The Kelly Criterion then determines optimal sizing: Kelly% = (p(odds-1) - (1-p)) / (odds-1).
Last reviewed: December 2025
Worked Examples
Example 1: Positive EV Bet Analysis
Example 2: Negative EV Bet Identification
Background & Theory
The Expected Value Betting Calculator applies the following established principles and formulas. Probability theory provides the mathematical foundation for analysing all games of chance. The fundamental measure assigns a probability between 0 and 1 to each outcome by dividing the count of favourable outcomes by the count of equally likely total outcomes. Rolling a standard six-sided die produces a 1/6 probability for each face; the probability that a fair coin lands heads exactly three times in five tosses follows the binomial distribution with parameters n=5 and p=0.5. Expected value (EV) is the probability-weighted average outcome of a random variable: EV equals the sum of each outcome multiplied by its probability. A fair coin flip paying $1 for heads and costing $1 for tails has EV of zero. Casino games are designed with negative expected value for the player; the house edge is the casino's average percentage profit per bet. European roulette with a single zero has a house edge of 2.7 percent, while American roulette's double zero raises it to 5.26 percent. Poker hand probabilities derive from combinatorics. From a 52-card deck, the number of distinct 5-card hands is C(52,5) = 2,598,960. A royal flush can occur in only 4 ways, giving it a probability of approximately 0.000154 percent. Blackjack basic strategy tables, derived from computer simulation of millions of hands, reduce the house edge from roughly 2 percent to below 0.5 percent by specifying the optimal hit, stand, double, or split decision for every player hand against every dealer up-card. Sports betting implied probability converts decimal odds to a probability estimate: implied probability equals 1 divided by decimal odds. Odds of 2.5 imply a 40 percent probability. The Kelly Criterion provides the theoretically optimal bet fraction: f equals (bp minus q) divided by b, where b is the net odds received, p is the probability of winning, and q is the probability of losing. This formula maximises the long-run geometric growth rate of a bankroll.
History
The history behind the Expected Value Betting Calculator traces back through the following developments. Physical evidence of dice play dates to around 2500 BCE at the Indus Valley city of Mohenjo-daro, where excavators found carved cubic astragali remarkably similar to modern dice. Ancient Egyptian, Greek, and Roman cultures all incorporated dice games into both leisure and religious ritual, suggesting gambling emerged independently across early civilisations as a universal human impulse. The first systematic attempt to mathematically analyse games of chance came from Gerolamo Cardano, the Italian polymath who wrote "Liber de Ludo Aleae" (Book on Games of Chance) around 1564. Cardano derived correct probabilities for dice combinations and introduced the concept of sample space, though his work remained unpublished until 1663. The field transformed into a rigorous discipline through correspondence in 1654 between Blaise Pascal and Pierre de Fermat prompted by a gambling problem posed by the Chevalier de Mere. Their exchange established the rules of probability, including the concept of expected value. Jacob Bernoulli's "Ars Conjectandi" (1713) formalised the law of large numbers, proving that sample frequencies converge to true probabilities as trials increase. The 20th century brought two pivotal developments. Stanislaw Ulam and John von Neumann devised Monte Carlo simulation methods in 1947 while working at Los Alamos, showing that complex probabilistic systems could be analysed by random sampling. Game theory and poker strategy developed in parallel, with John von Neumann's minimax theorem providing early foundations and later work by game theorists formalisingrational play under incomplete information. Online gambling launched in the mid-1990s following the passage of the Free Trade and Processing Act in Antigua in 1994, which issued the first online casino licences. The Unlawful Internet Gambling Enforcement Act of 2006 disrupted US online gambling markets. Esports betting and video game loot box mechanics brought probability and expected value calculations to younger audiences in the 2010s, prompting regulatory scrutiny of randomised virtual reward systems across multiple jurisdictions.
Frequently Asked Questions
Formula
EV = (Probability x Profit) - ((1 - Probability) x Stake)
Expected Value equals the probability of winning multiplied by the net profit if you win, minus the probability of losing multiplied by the amount you lose. A positive EV means the bet is profitable long-term. The Kelly Criterion then determines optimal sizing: Kelly% = (p(odds-1) - (1-p)) / (odds-1).
Worked Examples
Example 1: Positive EV Bet Analysis
Problem: You estimate a team has a 45% chance of winning. The sportsbook offers decimal odds of 2.50 ($100 stake). What is the expected value?
Solution: Profit if win = $100 x (2.50 - 1) = $150\nLoss if lose = $100\nEV = (0.45 x $150) - (0.55 x $100)\nEV = $67.50 - $55.00 = $12.50\nEV% = $12.50 / $100 = 12.5%\nImplied prob = 1/2.50 = 40%\nEdge = 45% - 40% = 5%\nKelly = (0.45 x 1.50 - 0.55) / 1.50 = 8.3% of bankroll
Result: EV: +$12.50 per bet (12.5%) | Edge: 5% | Kelly: 8.3%
Example 2: Negative EV Bet Identification
Problem: A coin flip game offers 1.91 decimal odds on heads. You know the true probability is 50%. Is this a good bet?
Solution: Profit if win = $100 x (1.91 - 1) = $91\nLoss if lose = $100\nEV = (0.50 x $91) - (0.50 x $100)\nEV = $45.50 - $50.00 = -$4.50\nEV% = -4.5%\nImplied prob = 1/1.91 = 52.4%\nEdge = 50% - 52.4% = -2.4%\nKelly = negative (do not bet)
Result: EV: -$4.50 per bet (-4.5%) | Negative edge | Do not bet
Frequently Asked Questions
What is expected value in sports betting and why does it matter?
Expected value (EV) is the average amount you can expect to win or lose per bet if you placed the same wager thousands of times. It is calculated by multiplying each possible outcome by its probability and summing the results. Positive EV means the bet is profitable in the long run, while negative EV means you will lose money over time. For example, if you have a $100 bet at 2.50 odds with a 45 percent true probability of winning, your EV is (0.45 x $150) - (0.55 x $100) = $67.50 - $55 = $12.50 per bet. This concept is the foundation of all professional sports betting strategies.
What is the Kelly Criterion and how does it relate to expected value?
The Kelly Criterion is a mathematical formula that determines the optimal bet size to maximize long-term bankroll growth when you have identified a positive expected value opportunity. The formula is: Kelly % = (probability x (odds - 1) - (1 - probability)) / (odds - 1). For example, with a 55 percent edge on a 2.00 odds bet, Kelly suggests betting (0.55 x 1 - 0.45) / 1 = 10 percent of your bankroll. Most professional bettors use fractional Kelly, typically half or quarter Kelly, to reduce variance and protect against estimation errors. Kelly only applies when EV is positive; when EV is negative, the formula returns zero or negative values.
Can a bet have positive expected value but still lose money?
Absolutely. Expected value is a long-term statistical concept, not a guarantee for any individual bet or even a series of bets. A bet with positive EV might lose in the short term because variance and randomness dominate small sample sizes. For example, a bet with 5 percent EV and a 40 percent win rate will experience losing streaks of 5 to 10 bets regularly. The law of large numbers means that actual results converge toward expected value only over hundreds or thousands of bets. This is why bankroll management is critical, as you need to survive the inevitable downswings to realize your long-term edge.
What is the difference between expected value and implied probability?
Implied probability is simply the probability suggested by the bookmaker odds, calculated as 1 divided by the decimal odds. Expected value compares your estimated true probability against the implied probability to determine if a bet offers value. If the implied probability from odds of 2.50 is 40 percent but you believe the true probability is 45 percent, you have found a positive EV situation with a 5 percent edge. The implied probability includes the bookmaker margin, so it is always slightly inflated. Understanding the gap between your estimated probability and the implied probability is the key to identifying profitable betting opportunities.
How many bets do I need to place before expected value is reliable?
The number of bets required depends on the size of your edge and the variance of your bets. As a general rule, you need at least 500 to 1,000 bets to start seeing a meaningful signal through the noise. With a 3 percent EV edge on standard -110 bets, you might need over 2,000 bets before your results become statistically significant. The higher your edge, the fewer bets you need. Professional bettors often track their closing line value instead of results in the short term because it provides a more immediate indicator of their skill. Statistical significance testing using p-values or confidence intervals can help determine when your results are unlikely to be due to luck alone.
Is it possible to consistently find positive expected value bets?
Yes, but it requires significant effort, skill, and discipline. Sharp bettors find positive EV through specialization in specific leagues or markets, building quantitative models, exploiting opening lines before they are adjusted, targeting soft bookmakers with less accurate odds, and line shopping across multiple sportsbooks. Steam moves and reverse line movement can also indicate value. However, finding consistent positive EV has become increasingly difficult as the market has become more efficient with the rise of algorithmic odds-making. Most successful bettors operate with thin margins of 2 to 5 percent EV and rely on high volume to generate meaningful profits.
References
Reviewed by Sher, Sports Science & Nutrition Specialist · Editorial policy