Probability and payout guide
How to Read Dice-Game Paytables
A paytable is the documented schedule of payouts for winning outcomes. To evaluate one correctly, you need more than the largest number on the table: you must identify the wager, named ruleset, payout basis, exceptions, outcome probabilities and amount used in the calculation.
- Editorial role
- Desk research and mathematical analysis
- Review status
- Outcome counts and expected-value examples checked
- Scope
- Educational analysis of dice-game paytables; not betting or legal advice
How the parts connect
- Wager rules Identify which outcomes win, lose or push.
- Outcome probabilities Show how often each result is expected to occur.
- Paytable entries Assign a net result to each category.
- Expected value Combines those probabilities and net results.
Text equivalent: wager rules define the outcome categories; probability determines the frequency of each category; the paytable assigns each category a net result; expected value combines the probabilities and net results.
If any link in this sequence is missing, the resulting house-edge claim cannot be reproduced reliably.
Before comparing two payouts: use the audit checklist on this page to confirm that both figures describe the same wager and rules.
On this page
Essential paytable and expected-value terms
- Paytable
- The documented schedule of payouts for winning outcomes. A useful paytable must be read together with the rules and exceptions governing the wager.
- Payout
- The amount assigned to a winning result. The source should state whether this means net profit or total return, including the original stake.
- Net winnings
- The profit from a winning settlement after excluding the returned stake. A net payout of 1 to 1 means one unit of profit for each unit wagered.
- Total return
- Everything returned after settlement, including the original stake. A one-unit stake with one unit of profit produces a two-unit total return.
- Stake
- The initial amount placed at risk on the wager being analyzed.
- Push
- A result that normally returns the stake without a win or loss. In a net-result calculation, an ordinary push is recorded as zero.
- Commission
- A fee charged under specified rules. A calculation must identify the amount, timing and basis of the charge—for example, whether it is collected when a wager is placed, wins or settles.
- Probability
- The likelihood of an outcome under stated assumptions. It can be expressed as an outcome count, fraction, decimal or percentage.
- Hit rate
- How often a wager produces a winning result. Hit rate does not account for the size of wins and losses, so a high hit rate does not necessarily imply favorable expected value.
- Expected value
- The probability-weighted average net result across all possible outcomes. A negative player expected value corresponds to a positive house advantage.
- House edge
- The average expected player loss as a percentage of the specified initial wager under a defined ruleset and paytable. It is not the probability of losing one round.
The fields every useful paytable should identify
A betting layout may show only the wager name and prominent payout. Important conditions—such as triple exclusions, ties, pushes, commissions or required underlying wagers—may appear only in the complete rules.
| Field | What it should tell you | Why it matters |
|---|---|---|
| Wager name | The exact bet being analyzed | Different wagers in the same game can have different probabilities and edges. |
| Named ruleset | The jurisdiction, venue rules, published version or documented variation | Paytables and exceptions are not necessarily universal. |
| Qualifying result | The roll or condition required to win | A label alone may not describe every qualifying combination. |
| Payout basis | The net payout or total-return convention | Confusing the two changes the expected-value calculation. |
| Excluded outcomes | Results that look eligible but lose under an exception | Sic Bo triples are a common example. |
| Push treatment | When the stake is returned without profit or loss | A push has a different net result from a loss. |
| Commission or fee | How much is charged and when | The fee can change expected value and the amount at risk. |
| Stake basis | The amount initially placed at risk on the wager, including any separately required or supplemental amount | The stake basis identifies what money is exposed and should not be assumed to be the same as the denominator used for a reported percentage. |
| Outcome counts | The number of winning, losing and pushing ordered outcomes | These counts make the probability calculation reproducible. |
| Calculation denominator | The disclosed amount against which the reported percentage is measured | This may differ from the complete amount at risk when attached wagers, combined wagers or fees are involved. |
| Source and version | The rule document, published version and verification date when available | Rules and published paytables can change. |
Table conclusion: a percentage is not sufficiently documented unless the reader can identify the wager, rules, settlement convention, stake basis, outcome counts, calculation denominator and source.
Text equivalent of the paytable anatomy
Start with a specific wager under a named ruleset. Identify every qualifying and excluded result, convert each payout to a net result, record pushes and fees, state the amount initially placed at risk, count the outcomes, separately state the calculation denominator, and preserve the source and document version. Do not combine rows drawn from different venue or jurisdictional paytables into an undocumented average.
Net payout versus total return
Net payout describes profit without counting the returned stake. Total return includes both the profit and the returned stake. The same settlement can therefore be described using two different numbers.
| Description | Stake | Profit | Amount returned | Net result used for expected value |
|---|---|---|---|---|
| Net payout: 1 to 1 | 1 unit | 1 unit | 2 units | +1 unit |
| Total return: 2 units | 1 unit | 1 unit | 2 units | +1 unit |
Table conclusion: “1 to 1 net” and “2 units returned for a one-unit stake” can describe the same settlement. The expected-value input is the consistent net result of +1 unit.
Unit-based settlement sequence
- Stake: one illustrative unit is placed at risk.
- Profit: the wager wins and produces one unit of profit.
- Stake returned: the original one-unit stake is also returned.
- Settlement: two units are returned in total, producing a net result of plus one unit.
Formatting alone does not establish which convention a source uses. Check the rules, then rewrite wins as positive net results, losses as negative net results and ordinary pushes as zero before calculating expected value.
Rules and exceptions that can change a payout calculation
Two prominent payout rows can look identical while producing different values because the underlying rules treat some outcomes differently. Ask the following questions before counting anything.
- Excluded outcomes: Does an apparently qualifying total lose because it is also a triple or another excluded combination?
- Pushes: Is any result settled by returning the stake rather than recording a win or loss?
- Commissions: Is a fee charged on the amount wagered, the win, or another basis—and at what point is it collected?
- Qualifying combinations: Does order matter, and must selected faces appear on specific dice?
- Top prizes: Has the payout for a rare result changed while the more common payouts remain the same?
- Attached wagers: Can the wager stand alone, or does it require a separate contract wager?
- Maximum odds: If a supplemental wager is offered, are its allowed amount and payout schedule limited by the rules?
Outcome-status legend
- Win: the settlement produces positive net winnings under the stated payout.
- Loss: the amount placed at risk is lost, subject to any separately stated fee or settlement rule.
- Push: the stake is returned without profit or loss, giving an ordinary net result of zero.
Mark each possible category explicitly as win, loss or push, then deduct any commission according to the rule governing when it is charged. For an example of point and odds rules that require a separate analysis, see the Crapless Craps rules and comparison.
Match each payout to its possible outcomes
An ordered outcome identifies the value shown by each die. With two dice, for example, a first die showing 2 and a second showing 5 is distinct from a first die showing 5 and a second showing 2, even though both totals equal 7.
Ordered-outcome denominators
- Two fair six-sided dice 6 possibilities for die one × 6 possibilities for die two 36 ordered outcomes
- Three fair six-sided dice 6 possibilities for die one × 6 for die two × 6 for die three 216 ordered outcomes
Text equivalent: two fair six-sided dice have 6 × 6 = 36 ordered outcomes. Three fair six-sided dice have 6 × 6 × 6 = 216 ordered outcomes.
Every ordered outcome must belong to one settlement category. If a calculation uses 216 possible outcomes, the winning, losing and pushing counts should add to 216. Missing or double-counted outcomes make the expected value unreliable.
For complete one-, two- and three-dice distributions, see Dice Probabilities, Expected Value and House Edge.
A step-by-step paytable audit
- Name the wager and ruleset. Record the exact bet, rule source and version or jurisdiction.
- Normalize the payouts. Convert each settlement to a net result measured in consistent units.
- List all outcome categories. Include wins, losses, pushes, exclusions and fee-triggering results.
- Count the ordered outcomes. Confirm that all categories add to the full denominator.
- Calculate each probability. Divide the category count by the total number of ordered outcomes.
- Weight each net result. Multiply each category’s probability by its net result.
- Sum the weighted results. The total is the player’s expected value per stated unit wagered.
- State the house-edge denominator. Identify whether the percentage is measured against the initial wager, a supplemental wager or another disclosed amount.
House edge is a theoretical long-run average under specified assumptions. It is not the chance of losing one round and cannot predict the next result.
Worked audit: Sic Bo Small with triples losing
This example analyzes a common Small wager: totals from 4 through 10 qualify, the wager pays 1:1 in net winnings, and all triples lose. The triple exception applies even when the triple’s total falls within the Small range.
| Wager | Small |
|---|---|
| Named rules assumption | Totals 4–10 win except all triples; even-money settlement |
| Stake basis | One unit initially placed at risk on the Small wager |
| Net payout | +1 unit on a win; −1 unit on a loss |
| Winning outcomes | 105 of 216 |
| Losing outcomes | 111 of 216 |
| Win probability | 105 ÷ 216, approximately 48.61% |
| Player expected value | (105 ÷ 216 × 1) + (111 ÷ 216 × −1) = −6 ÷ 216 |
| House-edge denominator | The one-unit initial Small wager |
| House edge | 6 ÷ 216, approximately 2.78% of the initial wager |
| Primary operational source | Singapore Gambling Regulatory Authority-hosted Sic Bo rules, used as a named rules reference rather than a universal paytable |
| Secondary mathematical check | Wizard of Odds Sic Bo analysis |
Table conclusion: the triple exception leaves 105 winning outcomes and 111 losing outcomes. The difference of six outcomes produces a player expected value of −6/216 and a house edge of approximately 2.78%.
Calculation
Assumptions: three fair six-sided dice, 216 equally likely ordered outcomes, net payout of +1 on a win, loss of −1 otherwise, and no pushes.
Formula: (105/216 × +1) + (111/216 × −1) = −6/216.
Result: player expected value is approximately −0.0278 units per unit initially wagered; house edge is approximately 2.78%.
See the complete Sic Bo rules, Small and Big assumptions, and venue-dependent paytables.
Worked comparison: changing the top Chuck-a-Luck payout
For a selected-number wager using three independent six-sided dice, the match distribution remains constant: 75 outcomes contain one match, 15 contain two matches, one contains three matches, and 125 contain no match.
| Outcome | Ordered outcomes | Schedule A net result | Schedule B net result |
|---|---|---|---|
| No match | 125 | −1 unit | −1 unit |
| One match | 75 | +1 unit | +1 unit |
| Two matches | 15 | +2 units | +2 units |
| Three matches | 1 | +3 units | +12 units |
| Player expected value | 216 total | (75 + 30 + 3 − 125) ÷ 216 = −17 ÷ 216 | (75 + 30 + 12 − 125) ÷ 216 = −8 ÷ 216 |
| House edge | Initial selected-number wager | 17 ÷ 216, approximately 7.87% | 8 ÷ 216, approximately 3.70% |
Table conclusion: the match probabilities are identical under both schedules. Raising only the three-match net payout from 3:1 to 12:1 changes the expected value and reduces the calculated house edge from approximately 7.87% to approximately 3.70%.
This comparison does not establish a universal Chuck-a-Luck paytable. It demonstrates why the exact payout assigned to every category must be recorded. Review the complete Chuck-a-Luck payout calculations or see how equivalent match probabilities require separate payout checks in the Crown and Anchor guide.
Attached wagers and the meaning of a 0% standalone edge
A craps odds wager illustrates why prerequisites and denominators matter. When paid at true mathematical odds, the supplemental odds wager has a standalone house edge of 0%. It cannot normally be placed by itself, however: it is attached to an underlying Pass, Come, Don’t Pass or Don’t Come contract wager.
Two-layer wager structure
- Required contract wager Establishes the underlying position and may carry a house edge.
- Supplemental odds wager Becomes available under the relevant rules and may be paid at true mathematical odds.
- Combined amount at risk Contains separate amounts with different mathematical treatment; any combined percentage must state its denominator.
Text equivalent: the contract wager is the required first layer. The true-odds wager is a supplemental second layer that depends on the contract wager. The complete amount at risk includes both layers, so a percentage for only the supplemental layer does not describe the combined proposition.
For the full relationship between a Pass Line wager and attached odds, read the Craps guide to Pass Line and true-odds wagers.
Red flags in payout and house-edge claims
- Missing wager: a percentage is assigned to an entire game rather than a particular bet.
- Missing ruleset: the source does not identify the jurisdiction, venue rulebook or documented variation.
- Missing paytable: the payout schedule behind the percentage is not shown.
- Missing formula or outcome counts: the claim cannot be reproduced from winning, losing and pushing results.
- Missing stake basis: the source does not identify the amount initially placed at risk.
- Missing denominator: the source does not state which amount is used to calculate the percentage.
- Lowest-edge substitution: one wager’s edge is presented as the edge for the whole game.
- Hit-rate substitution: a frequent win is presented as proof of favorable expected value.
- Omitted exceptions: triple rules, pushes, commissions or attached-wager requirements are absent.
- Mixed paytables: rows from different venues or jurisdictions are combined without separate calculations.
- No source or version: readers cannot identify which rules or paytable were checked.
- Past-roll prediction: the claim suggests that previous independent results make a future result due.
Random or properly generated outcomes do not make an unfavorable paytable favorable. The Fair Dice, RNGs and Game Integrity guide explains why randomness, payout value, licensing and profitability are separate questions.
Paytable audit checklist
- What exact wager is being analyzed?
- Which named ruleset, jurisdiction, venue rules or documented version applies?
- Are the listed payouts net winnings or total returns?
- Which outcomes win, lose or push?
- Do triples, ties, combinations or other exceptions override the apparent result?
- Is a commission or fee charged, and when?
- Does the wager require an underlying or attached wager?
- What amount is initially placed at risk?
- How many ordered outcomes belong to each settlement category?
- Do all outcome counts add to the full denominator—such as 36 for two dice or 216 for three?
- What net result is assigned to every category?
- What expected-value formula is used?
- Which amount is the denominator for the house-edge percentage?
- Are the source and applicable document version or verification date visible?
- Is a venue-specific paytable being incorrectly described as universal?
- Have variations been calculated separately rather than averaged without explanation?
Apply this sequence to the Sic Bo Small and Big analysis, the Chuck-a-Luck payout comparison, or the separate Crapless Craps ruleset analysis.
A paytable measures expected value, not what will happen next
Expected value is a theoretical long-run average under specified assumptions. It does not predict a single roll, a short session or the order in which outcomes will appear.
Calculation method, assumptions and sources checked
- Review status
- Definitions, ordered-outcome counts and displayed arithmetic checked through desk research and reproducible calculation.
- Calculation method
- Convert settlements to net unit results, count mutually exclusive ordered outcomes, multiply each result by its probability, sum the player expected value, and express house edge against the stated initial-wager denominator.
- General assumptions
- Fair independent six-sided dice unless otherwise stated; equally likely ordered outcomes; no commission, push or exception unless explicitly included.
- Sic Bo rules assumption
- Common Small rules covering totals 4 through 10, paying 1:1, with all triples losing. The regulator-hosted operational rules are treated as a named reference, not a universal paytable.
- Chuck-a-Luck assumptions
- One selected face on three independent six-sided dice, with separately analyzed net schedules of 1:1, 2:1 and 3:1 or 1:1, 2:1 and 12:1.
- Craps assumption
- The 0% statement applies only to a supplemental wager paid at true mathematical odds. It does not describe the required contract wager or the combined proposition.
Sources checked
- Singapore Gambling Regulatory Authority: Sic Bo Game Rules, Marina Bay Sands Version 5 (external PDF) — named operational rules source.
- Wizard of Odds: Sic Bo (external) — secondary mathematical check for outcome counts and wager-specific house edge.
- Wizard of Odds: Chuck-a-Luck (external) — secondary check for the match distribution and alternative payout calculations.
- Wizard of Odds: Craps Basics (external) — secondary reference for true-odds payouts and the required underlying contract wager.
Numerical claims are published only when the wager, ruleset, payout, outcome counts, formula, denominator and source can be identified. Read the Editorial Standards and Calculation Methodology for source selection and correction handling.
To report a mathematical, sourcing or accessibility issue, use the editorial contact options described on the About Dice Rules & Odds page.