Mathematics guide
Dice Probabilities, Expected Value and House Edge
Dice probability begins by counting all possible outcomes. For fair, independent six-sided dice, one die has 6 possible outcomes, two dice have 36 ordered outcomes, and three dice have 216 ordered outcomes. Those counts can be used to calculate probabilities, but a wager’s expected value also depends on its rules and paytable.
Probability describes what may happen over repeated trials. It cannot identify the result of the next independent roll. House edge is the expected player loss as a percentage of a specified initial wager—not the probability of losing one round.
- Editorial role
- Desk-researched educational reference
- Calculation scope
- One, two, and three fair six-sided dice under independent, unchanged rolling conditions
- Review status
- Outcome counts checked by complete enumeration and equivalent combinatorial calculations
- Publication record
- No last-modified or external-source verification date was supplied with the approved content. Date-dependent source claims remain identified in the methodology ledger.
Jump to the educational probability lookup or begin with ordered outcomes.
On this page
- Essential terms
- Counting ordered outcomes
- One-die probability
- Two-dice probability chart
- Three-dice totals and probabilities
- Educational probability lookup
- Independent rolls
- Hit rate, return, expected value, and house edge
- Expected-value formula
- Sic Bo worked calculation
- Interpreting house edge
- Calculation checklist
- Probability and gambling risk
- Methodology and sources
- Related reading
Essential probability and house-edge terms
- Ordered outcome
- A result that records the value shown by each individual die. With two dice, (2, 5) and (5, 2) are different ordered outcomes even though both total 7.
- Independent roll
- A roll whose probability distribution is unaffected by earlier rolls, assuming the dice are fair and the rolling conditions remain unchanged.
- Probability
- The proportion of the complete outcome space that satisfies a stated condition. For equally likely outcomes: probability = favorable ordered outcomes ÷ total ordered outcomes.
- Hit rate
- How frequently an event or wager produces a winning result. A high hit rate does not establish favorable expected value.
- Paytable
- The documented schedule of payouts for winning outcomes. A house-edge calculation is not valid without a defined paytable.
- Expected value
- The probability-weighted average net result of all possible outcomes. A negative player expected value corresponds to a positive house advantage when both use the same wager denominator.
- House edge
- The average expected player loss as a percentage of the specified initial wager under stated rules and payouts. It is not the probability of losing one round.
- Gambler’s fallacy
- The mistaken belief that earlier independent outcomes make an opposite or balancing result due on a later roll.
Net winnings and returned stake
A “1:1 net payout” means a one-unit win produces one unit of profit, with the original one-unit stake also returned. Some displays describe the total amount returned instead. Confusing net profit with total return changes an expected-value calculation. See how to audit payouts, returns, commissions, pushes, and exceptions.
Counting ordered outcomes for one, two, and three dice
Each fair six-sided die has six possible faces. When dice are distinguishable by position—such as first die, second die, and third die—the number of ordered outcomes is found by multiplying six once for each die.
- One die: 6 outcomes
- Two dice: 6 × 6 = 62 = 36 ordered outcomes
- Three dice: 6 × 6 × 6 = 63 = 216 ordered outcomes
For example, first die 2 and second die 5 is written as (2, 5). First die 5 and second die 2 is (5, 2). Both produce a sum of 7, but they occupy separate cells in the 36-outcome space.
General formula: Probability of an event = favorable ordered outcomes ÷ total ordered outcomes.
Text equivalent of the outcome-space diagram
A two-dice grid has six rows for the first die and six columns for the second die, producing 36 cells. Each cell represents one ordered pair. To calculate a probability, count every cell meeting the stated condition and divide that count by 36. The same method extends to 216 ordered triples for three dice.
One fair six-sided die
This table assumes a fair six-sided die and unchanged rolling conditions. Each face occupies one of the six equally likely outcomes.
| Face | Favorable outcomes | Total outcomes | Exact probability | Percentage |
|---|---|---|---|---|
| 1 | 1 | 6 | 1/6 | 16.67% |
| 2 | 1 | 6 | 1/6 | 16.67% |
| 3 | 1 | 6 | 1/6 | 16.67% |
| 4 | 1 | 6 | 1/6 | 16.67% |
| 5 | 1 | 6 | 1/6 | 16.67% |
| 6 | 1 | 6 | 1/6 | 16.67% |
Conclusion: Every face has the same theoretical probability, 1/6. A short series of observed rolls does not have to contain each face equally often, and this table does not predict the next result.
Two-dice probability chart
Two fair six-sided dice have 36 ordered outcomes. Central totals occur more often because more ordered pairs produce them. A total of 7 has six possible ordered pairs, while totals 2 and 12 each have only one.
| Total | Favorable ordered outcomes | Total ordered outcomes | Exact probability | Percentage |
|---|---|---|---|---|
| 2 | 1 | 36 | 1/36 | 2.78% |
| 3 | 2 | 36 | 2/36 = 1/18 | 5.56% |
| 4 | 3 | 36 | 3/36 = 1/12 | 8.33% |
| 5 | 4 | 36 | 4/36 = 1/9 | 11.11% |
| 6 | 5 | 36 | 5/36 | 13.89% |
| 7 | 6 | 36 | 6/36 = 1/6 | 16.67% |
| 8 | 5 | 36 | 5/36 | 13.89% |
| 9 | 4 | 36 | 4/36 = 1/9 | 11.11% |
| 10 | 3 | 36 | 3/36 = 1/12 | 8.33% |
| 11 | 2 | 36 | 2/36 = 1/18 | 5.56% |
| 12 | 1 | 36 | 1/36 | 2.78% |
Reproducible example: total 7
The ordered pairs producing 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). That is 6 favorable outcomes from 36:
6/36 = 1/6 ≈ 0.1667 = 16.67%
Conclusion: Frequencies rise to six outcomes for total 7 and then decrease symmetrically. Greater frequency does not make a result due or establish favorable expected value; payout and losing conditions also matter.
Three-dice totals and probabilities
Three fair six-sided dice have 216 ordered outcomes. Many different ordered triples can produce the same total, with totals 10 and 11 each occurring in 27 of the 216 outcomes.
| Total | Favorable ordered outcomes | Total ordered outcomes | Exact probability | Percentage |
|---|---|---|---|---|
| 3 | 1 | 216 | 1/216 | 0.46% |
| 4 | 3 | 216 | 1/72 | 1.39% |
| 5 | 6 | 216 | 1/36 | 2.78% |
| 6 | 10 | 216 | 5/108 | 4.63% |
| 7 | 15 | 216 | 5/72 | 6.94% |
| 8 | 21 | 216 | 7/72 | 9.72% |
| 9 | 25 | 216 | 25/216 | 11.57% |
| 10 | 27 | 216 | 1/8 | 12.50% |
| 11 | 27 | 216 | 1/8 | 12.50% |
| 12 | 25 | 216 | 25/216 | 11.57% |
| 13 | 21 | 216 | 7/72 | 9.72% |
| 14 | 15 | 216 | 5/72 | 6.94% |
| 15 | 10 | 216 | 5/108 | 4.63% |
| 16 | 6 | 216 | 1/36 | 2.78% |
| 17 | 3 | 216 | 1/72 | 1.39% |
| 18 | 1 | 216 | 1/216 | 0.46% |
Conclusion: Three-dice totals form a symmetric distribution, but frequency alone does not determine the value of a wager. A game may apply exclusions after totals are counted. For example, a Sic Bo range wager may treat a triple as a loss even when the triple’s total falls inside the range.
Sic Bo ranges, triple rules, and payouts vary by paytable. Read the complete Sic Bo rules and wager guide before applying these totals to a specific wager.
Educational dice probability lookup
This static, no-JavaScript lookup covers one, two, or three fair six-sided dice. Open the relevant panel, then find the target total or exact number of appearances. Each result reports favorable outcomes, total outcomes, a fraction, a percentage, and the method.
Assumptions: The dice are fair, rolls are independent, conditions remain unchanged, and every ordered outcome is equally likely. The outcome space is 6 raised to the number of dice. This lookup cannot predict the next roll and does not provide wagering, bankroll, staking, or profit advice.
Exact total with one die
Valid targets are 1 through 6. Every valid face has 1 favorable outcome from 6.
Result for any valid face: 1/6 = 0.1667 ≈ 16.67%.
Method: Count the one face matching the target and divide by the six equally likely faces.
Exact total with two dice
Valid targets are 2 through 12. Use the two-dice probability table to obtain the favorable count, simplified fraction, and percentage.
Example: Total 7 has 6 favorable ordered outcomes from 36, so 6/36 = 1/6 = 0.1667 ≈ 16.67%.
Method: Enumerate the 36 ordered pairs and count the pairs whose values add to the target.
Exact total with three dice
Valid targets are 3 through 18. Use the three-dice probability table to obtain the favorable count, simplified fraction, and percentage.
Example: Total 10 has 27 favorable ordered outcomes from 216, so 27/216 = 1/8 = 0.125 = 12.50%.
Method: Enumerate the 216 ordered triples and count the triples whose values add to the target.
Exact appearances of a selected face
The selected face may be any value from 1 through 6. Under the fair-die assumption, the result is the same for every selected face.
| Dice | Exact appearances | Favorable outcomes | Total outcomes | Simplified fraction | Percentage |
|---|---|---|---|---|---|
| 1 | 0 | 5 | 6 | 5/6 | 83.33% |
| 1 | 1 | 1 | 6 | 1/6 | 16.67% |
| 2 | 0 | 25 | 36 | 25/36 | 69.44% |
| 2 | 1 | 10 | 36 | 5/18 | 27.78% |
| 2 | 2 | 1 | 36 | 1/36 | 2.78% |
| 3 | 0 | 125 | 216 | 125/216 | 57.87% |
| 3 | 1 | 75 | 216 | 25/72 | 34.72% |
| 3 | 2 | 15 | 216 | 5/72 | 6.94% |
| 3 | 3 | 1 | 216 | 1/216 | 0.46% |
Method: For exactly k appearances among n dice, choose which k dice show the selected face; each remaining die has five nonmatching faces. Divide the resulting favorable count by 6n.
Verified test cases
| Input | Favorable outcomes | Total outcomes | Expected result |
|---|---|---|---|
| One die, exact total 4 | 1 | 6 | 1/6 = 16.67% |
| Two dice, exact total 7 | 6 | 36 | 1/6 = 16.67% |
| Three dice, exact total 10 | 27 | 216 | 1/8 = 12.50% |
| Three dice, exactly one appearance of face 6 | 75 | 216 | 25/72 = 34.72% |
Why previous rolls do not change the next independent roll
An independent roll starts with the same probability distribution regardless of earlier results, provided the dice are fair and the conditions have not changed. Rolling four sixes in succession does not make a non-six due. The next fair-die roll still gives face 6 a probability of 1/6.
The belief that a balancing result must arrive after a streak is the gambler’s fallacy. An unusual-looking sequence alone does not establish that physical dice or a digital system are unfair. Evidence about equipment, implementation, and statistical testing is a separate issue. See what physical-dice and RNG testing can establish.
Hit rate, return, expected value, and house edge are different measures
| Measure | What it answers | What it requires | What it does not show by itself |
|---|---|---|---|
| Hit rate | How often does the condition win? | Winning and total outcomes | The size of wins or losses |
| Return | What amount is paid or returned? | Clear payout wording, including whether stake is included | The average value across all outcomes |
| Expected value | What is the probability-weighted average net result? | Every outcome probability and net result | The result of one particular round |
| House edge | What is the expected player loss relative to the initial wager? | Named wager, ruleset, paytable, denominator, and expected value | The chance of losing one round or the order of future results |
Conclusion: A wager may win frequently yet still have negative expected value if its losses are larger than its wins or its payout is too low for the probability. The paytable guide explains returned stakes, net winnings, pushes, commissions, and exceptions.
How to calculate expected value
Expected value is the sum of each outcome’s probability multiplied by that outcome’s net result:
In plain language, multiply each possible net result by how often it occurs, then add those weighted results. Every outcome must be included exactly once.
If player expected value is −0.03 units per one unit initially wagered, the corresponding house advantage is 0.03 units, or 3%, when the same initial-wager denominator is used. Changing a payout, commission, push rule, or qualifying outcome changes the inputs and may change the result.
Worked example: Sic Bo Small or Big at 1:1 with triples losing
This example uses the commonly analyzed structure in which Small covers totals 4 through 10 and Big covers totals 11 through 17, but all triples lose. A winning wager pays 1:1 in net winnings. The exact range wording, triple exception, and payout must be checked in the applicable rules and paytable.
| Wager | Small or Big |
|---|---|
| Named ruleset | Small 4–10 or Big 11–17; all triples lose |
| Net payout | 1:1 |
| Winning outcomes | 105 of 216 |
| Losing outcomes | 111 of 216 |
| Hit rate | 105/216 ≈ 48.61% |
| Player expected value | −6/216 ≈ −0.02778 units per unit initially wagered |
| House edge | 6/216 = 1/36 ≈ 2.78% |
| Calculation denominator | One unit initially wagered |
| Primary rules support | Singapore Gambling Regulatory Authority-hosted Sic Bo rules named in the source ledger |
| Secondary check | Direct enumeration and the specialist mathematical reference named below |
| Verification date | Not supplied in the approved research record; publication verification remains required |
Step-by-step calculation
- Three dice produce 63 = 216 ordered outcomes.
- Under the stated wager rules, 105 outcomes win and 111 outcomes lose.
- The hit rate is 105 ÷ 216 ≈ 48.61%.
- A win has a net result of +1 unit; a loss has a net result of −1 unit.
- Player EV = ((105 × 1) + (111 × −1)) ÷ 216.
- Player EV = (105 − 111) ÷ 216 = −6/216 ≈ −0.02778 units.
- The corresponding house edge is 6/216 = 1/36 ≈ 2.78% of the initial wager.
The hit rate, 105/216, is not the house edge. Hit rate counts how often the wager wins; house edge measures the average expected cost after the payout and every losing outcome are included.
For round flow and other wager categories, read the complete Sic Bo guide.
What house edge does—and does not—tell you
House edge does describe
- The theoretical average expected cost of a specified wager under stated rules.
- A percentage based on a defined denominator, usually the initial amount wagered on this site.
- The mathematical effect of the complete paytable over repeated trials.
House edge does not describe
- The probability of losing one round.
- The order in which wins and losses will occur.
- A guaranteed result for an individual session.
- The edge of every wager in the same game.
- Whether a game is lawful or available in a particular jurisdiction.
Individual results can differ substantially from the theoretical average. Any change to a payout, commission, push rule, qualifying outcome, or calculation denominator requires a new calculation.
Before accepting any percentage, use the paytable-audit guide to check its payout assumptions and exceptions.
Checklist for verifying a dice probability or house-edge claim
- Identify the exact event or wager. Do not accept a percentage described only as “the game’s edge.”
- Name the ruleset. Record the jurisdiction, venue rulebook, historical version, or other documented source.
- Record the paytable. Include net payouts, returned stakes, commissions, pushes, and exceptions.
- Count the outcome space. State the total number of equally likely ordered outcomes.
- Count favorable and losing outcomes. Apply exclusions explicitly.
- Define net results. Use consistent positive, negative, and zero values for wins, losses, and pushes.
- State the denominator. Explain whether the percentage is relative to the initial wager or another amount.
- Show the formula. Readers should be able to reproduce the result.
- Cite the source and verification date. Rule or payout changes can invalidate an older calculation.
- Keep wager and game claims separate. One wager’s edge must not be presented as the edge of the entire game.
For a detailed audit process, use How to Read Dice-Game Paytables. For source selection and review practices, see Editorial Standards and Calculation Methodology.
Probability cannot predict the next roll or create a recovery opportunity
No probability table, calculator, pattern, or recent sequence can predict the next independent roll. Prior losses do not improve the probability of recovering money on a later roll, and chasing losses increases risk.
If gambling is causing concern for you or someone close to you, free and confidential support is available through the National Council on Problem Gambling. Call or text 1-800-MY-RESET. NCPG describes the national helpline as available 24 hours a day, seven days a week. Other access points or jurisdiction-required numbers may also remain in use.
Calculation method, assumptions, and sources checked
Methodology ledger
- Base assumptions
- Fair six-sided dice; rolls are independent; rolling conditions remain unchanged; each ordered outcome is equally likely.
- One-die table
- Direct enumeration of six outcomes.
- Two-dice table
- Complete enumeration of 36 ordered pairs, checked against the symmetric combinatorial distribution.
- Three-dice table
- Complete enumeration of 216 ordered triples, checked against counts 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, and 1.
- Sic Bo example
- Small 4–10 or Big 11–17, all triples losing, 1:1 net payout, 105 winning outcomes, and 111 losing outcomes.
- Calculation convention
- Expected value uses net wins and losses. House edge is expressed relative to one unit initially wagered.
- Lookup edition
- Static lookup edition 1.0, designed to work without JavaScript and checked against the published one-, two-, and three-dice tables.
- External-source check date
- Not supplied in the approved research record. Current rules and links require editorial verification before being described as currently checked.
- Editorial role
- Desk research and calculation review; no hands-on casino testing or unsupported expert credential is claimed.
Sources
- Singapore Gambling Regulatory Authority-hosted Sic Bo Game Rules, Marina Bay Sands Version 5 — regulator-hosted game rules named as the operational rules reference.
- Wizard of Odds: Sic Bo — specialist mathematical reference named as a secondary check for outcome counts and wager analysis.
- Direct combinatorial calculation — 6 × 6 = 36 ordered outcomes for two dice and 6 × 6 × 6 = 216 for three dice.
Correction history
- Completeness repair: replaced the nonfunctional scripted calculator with a complete static lookup, added the missing two-dice visualization, and exposed a direct correction form.
- No earlier meaningful corrections were included in the supplied research record.
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