Expected Value

Expected value, usually written EV, is the probability-weighted average of all possible net results of a decision. It answers a forward-looking question: what is this decision worth on average before the random outcome 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 for that outcome;
• the outcomes cover all possibilities without overlap;
• 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 practical costs such as fees, taxes, time, liquidity, or execution errors.

EV belongs to the decision, not the result that happened once

A negative-EV bet can win today. A positive-EV opportunity can lose today. Neither result changes the value the decision had when it was made.

That separates three ideas:

• result: what happened this time;
• expected value: the probability-weighted center of the decision;
• variance: how widely individual outcomes can spread around that center.

Judging the quality of a wager only by whether it won is outcome bias. EV evaluates the price of the decision using information available before the outcome is selected.

A complete roulette EV calculation

Take a one-unit straight-up bet on one number in double-zero American roulette. There are 38 pockets. One wins and 37 lose. The winning bet pays 35 to 1, so the net result is +35 units on a win and −1 unit on a loss.

[ EV=\left(\frac{1}{38}\times35\right)+\left(\frac{37}{38}\times-1\right) ]

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

The wager therefore has expected value of about −0.05263 unit per unit staked.

For a fixed one-unit casino wager using the same denominator:

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

so:

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

That does not predict the next spin or a particular session. It describes the long-run center of repeated identical decisions.

Probability and payout have to be combined

Hit frequency alone is not value. Headline payout alone is not value. EV combines both.

For a simple one-unit wager with win probability (p) and net win of (R) units:

[ EV=pR-(1-p) ]

The fair net payout makes EV equal to zero:

[ 0=pR-(1-p) ]

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

If an outcome has a 10% chance, the fair net payout is:

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

So 9 to 1 is fair before other costs. If the bet pays only 8 to 1:

[ EV=(0.10\times8)-(0.90\times1)=-0.10 ]

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

EV, house edge, RTP, expected loss, and variance answer different questions

For compatible fixed-stake 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\times0.02=$100 ]

The expected $100 loss is not a forecast that the session will finish exactly −$100. Variance determines how dispersed the actual result can be.

State the denominator before comparing percentages

EV can be expressed per initial wager, per total amount committed, per hand, per spin, per hour, or per complete strategy sequence. Percentages using different denominators should not be compared as though they describe the same thing.

This is especially important in games with follow-up wagers. A carnival game might quote an edge relative to the Ante, while another source quotes average loss relative to total action after raises are included. Both calculations can be internally correct while answering different questions.

A reproducible EV statement identifies:

• the exact game and wager;
• the rules or paytable;
• the strategy assumptions;
• whether payouts are net profit or total return;
• the denominator;
• any commission, fee, push, surrender, cap, or required additional wager.

Decision EV can rank choices even when every choice is negative

In blackjack, video poker, or poker-style casino games, the best available action does not have to be positive EV. It can simply be less negative than the alternatives.

Suppose three legal actions have EVs of:

• Action A: −0.40 unit
• Action B: −0.15 unit
• Action C: −0.28 unit

Action B is the best decision even though it still loses 0.15 unit on average. Good strategy often means minimizing a disadvantage rather than turning the whole game positive.

This is why a correct play can lose immediately while a bad play wins. The outcome does not retroactively change the ranking of the decisions.

Conditional EV helps with branching decisions

Some casino decisions occur after new information appears. In that case, EV should be calculated using the probabilities that apply at that decision point, not the probabilities that existed before the new information was known.

For example, a video-poker hold decision is evaluated using the cards already dealt and the unseen-card distribution. A blackjack hit/stand decision is evaluated using the player’s current hand, dealer information, and the rules in force.

This is conditional expected value: the calculation is updated after relevant information becomes known.

The same principle prevents a common mistake—using an overall game average to evaluate a specific decision that has a different conditional distribution.

Expected values add across multiple decisions

Linearity of expectation allows separate EVs to be added:

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

If every wager is identical:

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

This remains useful even when outcomes are not independent. Independence matters for many probability and variance calculations, but not for the basic rule that expected values add.

That is why total expected loss can be estimated by adding the average cost of many separate wagers while the actual path remains uneven.

Positive EV does not mean low risk

A positive-expectation opportunity can still have severe short-term losses. A negative-expectation game can still produce a large short-term win.

The sign of EV says where the long-run average points. It does not tell you:

• how volatile the path is;
• how much bankroll is required;
• whether the opportunity can be repeated;
• whether assumptions can change;
• whether execution errors will erase the edge.

That is why variance belongs beside EV in any serious risk analysis.

Calculation errors usually come from definitions, not arithmetic

The most common EV mistakes are:

• using total return as though it were net profit;
• omitting one or more losing outcomes;
• using probabilities that do not sum to 1;
• mixing a probability from one rule set with a payout from another;
• treating a push as a win or a loss;
• ignoring commission or an additional required wager;
• comparing EV per hand with EV per dollar;
• using an unconditional average for a conditional decision;
• assuming a short sample should equal the theoretical average.

A strong calculation lets another reader rebuild the result from the stated rules and inputs.

For a broader statistical treatment, the NIST probability and expected-value tutorial uses the same probability-weighted-average principle outside casino games.

EV tells you the average price, not the full experience

Expected value is powerful because it puts probability and payoff on the same scale. It can compare decisions, estimate average cost, and identify whether a price is favorable or unfavorable under stated assumptions.

It cannot tell you the next outcome or describe the full distribution by itself.

Use probability and payout odds to build the inputs, variance to understand dispersion, and the expected-loss calculator to translate a house edge and planned amount of action into money.

The compact rule is: probability tells you how often, payout tells you how much, and expected value combines the two into the average price of the decision.