Expected Value

Expected value, usually written as EV, is the probability-weighted average of all possible net results of a decision. It measures what the decision is worth on average before the actual result is known.

The general formula is:

[ EV = \sum_{i=1}^{k} p_i x_i ]

where:

• (k) is the number of possible outcomes;
• (p_i) is the probability of outcome (i);
• (x_i) is the net gain or loss if outcome (i) occurs;
• the outcomes are mutually exclusive and collectively exhaustive;
• the probabilities sum to 1.

A positive EV favors the decision-maker on average. A negative EV costs the decision-maker on average. Zero EV is mathematically fair before considering fees, time, liquidity, taxes, or other practical costs.

EV evaluates the decision, not the result

A negative-EV bet can win. A positive-EV opportunity can lose. One outcome does not change the value the decision had when it was made.

This distinction is central to casino math:

• result: what happened this time;
• expected value: the average result implied by the probabilities and payoffs;
• variance: how widely individual results can spread around that average.

Judging a wager only by whether it won is outcome bias. EV asks whether the price was favorable before luck selected the outcome.

A complete roulette calculation

Consider a one-unit straight-up bet on one number in double-zero American roulette. The wheel has 38 pockets. One pocket wins and 37 lose. The winning wager pays 35 to 1, meaning the net result is +35 units on a win and −1 unit on a loss.

[ EV = \left(\frac{1}{38} \times 35\right) + \left(\frac{37}{38} \times -1\right) ]

[ EV = \frac{35-37}{38} = -\frac{2}{38} \approx -0.0526316 ]

The expected value is approximately −0.05263 unit per unit wagered. In money terms, a $1 wager has an average value of about −5.263 cents under these rules.

For a fixed one-unit casino wager, the house edge is the negative of player EV expressed as a percentage of the initial stake:

[ \text{House edge} = -\frac{EV}{\text{stake}} ]

Here:

[ -\frac{-0.0526316}{1} = 0.0526316 = 5.26316% ]

The calculation does not predict the next spin or any particular session. It describes the long-run center of repeated identical wagers.

Why probability and payout must be combined

A prize can be large and still have poor expected value because it is too rare. A bet can win frequently and still have poor expected value because each loss is too expensive. Neither hit frequency nor headline payout is enough by itself.

For a simple wager with probability (p) of winning and net payout (R) units per unit staked, the expected value is:

[ EV = pR - (1-p) ]

The fair net payout sets EV to zero:

[ 0 = pR - (1-p) ]

[ R = \frac{1-p}{p} ]

Suppose an outcome has a 1-in-10 chance, so (p=0.10). Its fair net payout is:

[ R = \frac{0.90}{0.10} = 9 ]

A fair wager would pay 9 to 1. If it pays 8 to 1, then:

[ EV = (0.10 \times 8) - (0.90 \times 1) = -0.10 ]

The player gives up an average of 0.10 unit per unit wagered.

Expected value, house edge, RTP, and expected loss

These terms are related but not interchangeable.

For a simple fixed-stake wager with compatible definitions:

[ \text{Expected loss} = \text{total action} \times \text{house edge} ]

If 500 identical $10 bets have a 2% house edge, total action is $5,000 and expected loss is:

[ $5{,}000 \times 0.02 = $100 ]

Linearity of expectation allows the average values of multiple wagers to be added even when outcomes are not independent:

[ EV_{\text{total}} = \sum_{j=1}^{n} EV_j ]

If every wager has the same stake and EV, this simplifies to:

[ EV_{\text{total}} = n \times EV_{\text{per wager}} ]

That property does not make actual results smooth. It tells us the average of the combined exposure.

The denominator must be stated

Expected value may be expressed per initial wager, per total amount committed, per hand, per spin, or per complete strategy decision. Confusion begins when percentages using different denominators are compared as though they describe the same thing.

This is especially relevant in games where a player may add wagers after the first decision. A percentage quoted against the ante alone can differ from a percentage quoted against average total action. Both may be mathematically valid while answering different questions.

A complete EV statement identifies:

• the exact game and wager;
• the rules or paytable;
• the strategy assumptions, if decisions affect probabilities or stakes;
• whether payouts are net profit or total return;
• the denominator;
• any commission, fee, push, surrender, or bonus rule.

Decision EV versus game EV

“Game EV” is often shorthand for the expectation produced by a complete set of rules and an assumed strategy. “Decision EV” compares available actions at one decision point.

In blackjack or video poker, the best available action can still have negative EV. It is best because it loses less, or wins more, on average than the alternatives. Strategy does not need to turn the whole game positive to improve a decision.

This is why a correct play can lose immediately and an incorrect play can win. The result does not retroactively change the probabilities.

Positive and negative expectation

A standard casino wager is usually a negative-expectation proposition for the player. Some promotions, skill-dependent situations, pricing errors, or advantage-play conditions may create positive expectation, but positive EV still carries risk and often depends on assumptions that can change.

The mathematical sign alone does not answer practical questions about bankroll, variance, time, execution errors, account restrictions, taxes, or whether the stated opportunity can be repeated.

For a non-casino treatment of probability-weighted averages, the NIST probability and expected-value tutorial provides the same underlying principle in a broader statistical setting.

Common calculation errors

The most frequent EV mistakes are:

• using the total amount returned as though it were net profit;
• leaving out losing outcomes;
• assigning probabilities that do not sum to 1;
• using a probability from one rule set with a payout from another;
• treating a push as a win or loss;
• ignoring commission or additional required wagers;
• comparing EV per hand with EV per dollar;
• assuming a short sample should equal the theoretical average;
• changing stakes and then discussing “the EV” without stating the new exposure.

A reproducible calculation should let another reader rebuild the result from the rules.

What EV can and cannot tell you

Expected value can identify the average mathematical price of a decision, compare available choices, and estimate the center of repeated exposure. It cannot promise the next result, describe the entire distribution, or tell a player how much bankroll is sufficient by itself.

Use probability and payout odds to build EV, variance to understand the spread, and the expected-loss calculator to translate an edge and planned action into money.

The concise rule is: probability tells you how often, payout tells you how much, and expected value combines the two.