Casino math becomes much easier to audit when the calculation is separated into a few explicit steps. Chips & Truths does not treat “odds,” “RTP,” “house edge,” “hit frequency,” and “hold” as interchangeable casino words. Each describes a different quantity.
This page explains the calculation method used across the site. The broader evidence policy is on How Chips & Truths Uses Casino Data and Evidence.
Step 1: define the exact wager and ruleset
The calculation begins before any arithmetic.
A useful game definition may need to specify:
- wheel type;
- number of decks;
- dealer drawing rule;
- blackjack payout;
- double and split restrictions;
- baccarat commission rule;
- side-bet paytable;
- video-poker paytable;
- joker or wild-card rules;
- progressive or bonus contribution;
- player-strategy assumption.
Without that definition, a precise-looking percentage can answer the wrong game.
For example, “roulette house edge” is incomplete if the page does not say whether the wheel has one zero or two. “Blackjack house edge” is incomplete if one page assumes 3:2 blackjack and another silently assumes 6:5. “Video poker RTP” is incomplete without the paytable and strategy standard.
Step 2: list the possible outcomes
For a simple wager, the outcomes may be only win and lose. Other bets can include multiple winning categories with different payouts.
The core requirement is that the outcome set is complete. If an event can push, pay half, trigger a bonus, lose only part of the wager, or carry into another state, that outcome belongs in the calculation.
A basic expected-value model is:
[ EV=\sum_i P_i \times X_i ]
where:
- (P_i) is the probability of outcome (i);
- (X_i) is the net win or loss for that outcome.
The probabilities across mutually exclusive outcomes should add to 1, subject only to rounding.
Step 3: separate probability from payout
Probability describes how often an outcome occurs under the model. Payout describes what the casino returns when it occurs.
Those are connected but not the same.
Roulette single-number example
On a single-zero roulette wheel there are 37 pockets. A straight-up number therefore has probability:
[ P(\text{win})=\frac{1}{37}\approx2.7027% ]
The casino payout is normally 35 to 1, meaning a $1 winning wager produces $35 profit plus return of the original $1 stake.
The net expected value per $1 wager is:
[ EV=\frac{1}{37}(+35)+\frac{36}{37}(-1) ]
[ EV=-\frac{1}{37}\approx-0.027027 ]
So the house edge is about 2.70%.
The important point is not memorizing the decimal. It is seeing where it comes from: 37 possible pockets, a 35-to-1 payout, and one winning pocket.
Step 4: convert expected value into house edge when appropriate
For a conventional wager where the player risks one unit, house edge can be expressed as the negative expected value divided by the amount initially wagered:
[ \text{House Edge}=\frac{-EV}{\text{Initial Wager}} ]
If the expected value is -$0.027 per $1 wager, the house edge is about 2.7%.
This is a long-run average price of action. It does not predict what one player will lose in one session.
Step 5: connect house edge to expected loss without pretending it predicts the session
A common explanatory formula is:
[ \text{Expected Loss}=\text{Total Action}\times\text{House Edge} ]
Suppose a player makes 100 wagers of $10 on a bet with a 2% house edge:
[ \text{Total Action}=100\times$10=$1{,}000 ]
[ \text{Expected Loss}=$1{,}000\times0.02=$20 ]
That does not mean the player should expect to finish exactly $20 down. It means that across enough equivalent action, the average result is centered around that theoretical cost. Variance controls how widely individual sessions can move around the average.
Step 6: convert between RTP and house edge only when the definitions match
For many fixed casino wagers, the relationship is:
[ \text{RTP}=1-\text{House Edge} ]
A 1% house edge corresponds to a 99% theoretical return under the same wager definition.
But this shortcut should not be used carelessly. Some products quote return using rules, bonuses, jackpots, or strategy assumptions that must be understood first. Casino accounting “hold” is also not simply the inverse of RTP.
True odds and payout odds answer different questions
True odds describe the fair relationship implied by probability. Payout odds describe what the game actually pays.
If an event has a 1-in-6 probability, fair profit odds would be 5 to 1. If a casino pays only 4 to 1 on that event, the missing value contributes to the house advantage.
The site uses this comparison frequently because it makes the price of a wager visible without requiring advanced notation.
Multi-outcome wagers require every paytable line
Side bets and video poker cannot be judged from the headline jackpot alone.
Imagine a simplified $1 side bet with three winning outcomes:
| Outcome | Probability | Net payout |
|---|---|---|
| Rare premium hand | 0.5% | +$40 |
| Common winning hand | 10% | +$3 |
| All other outcomes | 89.5% | -$1 |
The expected value is the probability-weighted sum of all three lines. A spectacular top prize can coexist with a poor overall return when most of the distribution is weak.
This is why Chips & Truths pages repeatedly tell readers to inspect the whole paytable instead of the largest printed number.
Blackjack needs a strategy assumption
Blackjack is different from roulette because the player makes decisions. The value of a ruleset depends partly on whether the player hits, stands, doubles, splits, and surrenders correctly.
A blackjack figure should therefore identify whether it assumes:
- basic strategy;
- a specific rule set;
- composition-dependent play;
- an advantage-play method;
- or average/error-prone play.
A page should not present a basic-strategy house edge and then imply that every player automatically receives that result.
Video poker needs a paytable and a hold strategy
Video poker also combines fixed paytable mathematics with player decisions. Two machines carrying the same game name can have different returns if the full-house, flush, four-of-a-kind, wild-card, or royal-flush lines differ.
The strategy used to choose which cards to hold matters as well. A theoretical RTP based on optimal play should not be presented as the return of random holds.
Slots often require published or technical return information
A reader normally cannot reconstruct a modern slot’s exact RTP by watching a short sample of spins. The visible reel result does not reveal every internal symbol weight, bonus probability, state, feature contribution, or jackpot component.
For those games, the site distinguishes between:
- published RTP or game information;
- visible paytable rules;
- observable session outcomes;
- and claims that cannot be derived from the available data.
A run of 100 losing spins cannot prove the long-run RTP is false, just as one jackpot cannot prove it is generous.
Rounding is done after the important arithmetic
Premature rounding can create small inconsistencies, especially when several outcome categories are summed. The preferred approach is to keep sufficient precision through the calculation and round the reader-facing result at the end.
The number of displayed decimal places should match the decision. A comparison between 2.70% and 5.26% does not become more useful because each is printed to eight decimals.
Sanity checks catch many calculation errors
Before publishing a derived casino number, several quick tests help:
- Do mutually exclusive probabilities sum to approximately 100%?
- Is the sign of the expected value sensible?
- Does a worse payout make the player return worse, not better?
- Does adding a player-friendly rule move the result in the expected direction?
- Does the formula use net profit rather than accidentally counting returned stake twice?
- Does the result agree reasonably with an independent reference or second calculation when one exists?
- Is the result tied to the exact rules stated on the page?
If a result fails one of those checks, the calculation should be re-examined before the decimal is treated as authoritative.
Worked examples are designed to be replaceable
A good example lets the reader substitute different inputs.
For hourly expected loss:
[ \text{Hourly Action}=\text{Average Wager}\times\text{Decisions per Hour} ]
[ \text{Hourly Expected Loss}=\text{Hourly Action}\times\text{House Edge} ]
If a player bets $25 instead of $10, the expected-cost example scales. If the game slows from 80 decisions per hour to 40, the action falls. The formula reveals which variable is responsible.
That is more useful than giving one unexplained “average loss” number and asking the reader to trust it.
What a calculation can and cannot tell you
Casino mathematics can tell you the long-run price of a defined wager, compare rules, identify dominated bets, estimate theoretical loss, and show how pace and bet size change exposure.
It cannot tell you:
- which independent roulette number will appear next;
- when a random jackpot is “due”;
- whether a short session will finish up or down;
- whether a player can emotionally tolerate the variance;
- whether a local casino is using a rule that has not been verified;
- whether a proprietary game has hidden probabilities that were never disclosed.
The calculation is strongest when its limits are stated as clearly as its answer.
The publication standard
A casino-math page should make it possible for a careful reader to identify:
rules → outcomes → probabilities → payouts → expected value → practical meaning.
When one of those links is unavailable, the page should say so rather than manufacture certainty.
For the broader evidence process, continue with How Chips & Truths Uses Casino Data and Evidence. For the site-wide writing standard, read Methodology and Editorial Principles.