Expected Value in Betting

Most bettors confuse expected value in betting with the chance of winning a single bet. A bet can have a 40% chance of winning but a negative EV, while a 1-in-10 shot can be positive EV if the payout is high enough. EV is the long-run average profit or loss per dollar wagered, calculated as (probability of win × profit) − (probability of loss × stake). It tells you nothing about the next spin, but everything about what happens after a thousand spins. OpenIntro Statistics (4th ed., section 3.4) defines it as E(X) = Σ xᵢ · P(X = xᵢ).

For a bet that pays $W on a win and costs $1, with win probability p: EV = (p × W) − ((1 − p) × 1). A casino bet that pays 35 to 1 on a single roulette number has p = 1/38 (double-zero wheel). EV = (1/38 × 35) − (37/38 × 1) = −0.0526, or −5.26 cents per dollar. Blitzstein & Hwang (2nd ed., ch. 4) emphasise that this formula assumes the bettor knows the true probability, which is rarely the case outside casino games.

To work out whether a bet is +EV, you must convert the offered odds into an implied probability. The table below shows the conversion for American, decimal and fractional odds.

Bookmakers never offer odds that sum to 100% implied probability. They add a margin called the vig or overround. On a two-outcome market (e.g., a coin flip), fair odds would be +100 each, each implying 50% probability, summing to 100%.76%. That extra 4.76% is the bookmaker's profit margin. The sum of implied probabilities always exceeds 100% in a market with a vig.

The house edge is the casino's long-run profit per dollar wagered, equal to the negative of the player's EV. Wizard of Odds publishes house-edge tables for casino games. The values vary by game and rule set.

Roulette Worked Example

On a double-zero American roulette wheel (38 slots), a single-number bet pays 35 to 1. The true probability of winning is 1/38. EV per $1 bet = (1/38 × $35) − (37/38 × $1) = −$0.0526. The house edge is 5.26%. On a single-zero European wheel (37 slots), the same bet pays 35 to 1, probability 1/37, EV = (1/37 × $35) − (36/37 × $1) = −$0.0270, a house edge of 2.70%.

Other casino games have house edges published by Wizard of Odds: blackjack with basic strategy ≈ 0.5% (rising with player errors), baccarat ≈ 1.06% on the banker bet, and craps pass line bet ≈ 1.41%. Slot machine RTP is set by regulators but typically 85-95%, giving a house edge of 5-15%.

A positive EV bet occurs when the true probability of an event is higher than the implied probability from the odds. This is rare in casino games because the house edge is built into the rules. In sports betting, positive EV can arise when a bookmaker misprices a market, or when a bettor has better information (e.g., an injury that the market has not adjusted for). In poker, a player with a skill edge can have +EV against weaker opponents. Kelly Criterion (1956) provides the optimal fraction of bankroll to bet on a +EV opportunity: f* = (bp − q) / b, where b is the net odds, p is the win probability, and q is the loss probability. Betting more than the Kelly fraction risks ruin despite positive EV.

Positive EV bets are fleeting. In efficient sports markets, mispricing is corrected within hours. In casino games, no bet has positive EV over the long run except through advantage play (e.g., card counting in blackjack, which is not allowed). The lottery is a special case: advertised jackpots can make the raw EV appear positive, but after accounting for split risk (multiple winners) and taxes (IRS 24% withholding on large prizes in the US), the after-tax EV is almost always negative.

EV is a long-run average; variance means you can lose a +EV bet 20 times in a row. A positive EV does not guarantee a win on the next play. The UK Gambling Commission and NCPG (National Council on Problem Gambling) provide resources for setting limits: never gamble money you cannot afford to lose, set a loss limit before you start, and walk away when you hit it. The NCPG resource page includes a self-assessment for problem gambling. The UK Gambling Commission's responsible gambling page sets standards for operator player protection. For US players, the IRS requires payers to issue Form W-2G for certain gambling winnings; these are taxable as ordinary income.

EV betting suits anyone who wants to evaluate a wager over many trials rather than chase a single win. It is the right tool for a student working through probability homework, a bettor comparing two sports markets, or a project manager using Expected Monetary Value (EMV) in a decision tree. It does not suit anyone looking for a guarantee on the next bet. If you need to know what happens on one roll of the dice, use a probability-of-an-event calculator. If you are an investor, EV alone is insufficient without considering variance, utility and risk tolerance, as Blitzstein & Hwang (ch. 4, section 4.8) note in their expected utility framework.

The most common failure: assuming a +EV bet is a sure thing. It is not. Variance and bankroll ruin are real. A bettor who overbets a +EV opportunity by ignoring the Kelly Criterion will go bust before the law of large numbers kicks in.