Advanced reference · Exact fractions · Worked proofs
This workbook shows how craps probabilities, fair odds, expected value, and house edge are calculated from the 36 ordered outcomes of two dice. It derives the major line, Place, Buy, Lay, Field, Horn, Hardway, and proposition-bet figures instead of asking you to trust a chart.
What the math can—and cannot—do
The calculations can tell you the price of a wager, expose a bad payout, and show how much additional Odds changes your total risk. They cannot predict the next independent roll or turn a negative-expectation bet into a winning system. Use the page as a verification reference, not as a promise that enough algebra beats the game.
Scope and assumptions
- Two balanced six-sided dice are rolled, producing 36 equally likely ordered outcomes.
- Standard bar-12 Don’t Pass rules are used unless a variation is named.
- “Net profit” excludes the returned wager. A payout written “7 to 6” means $7 profit for each $6 wagered.
- Buy and Lay commission timing is stated in each example; win-only and upfront commission produce different results.
- House edge is normally expected loss divided by the initial wager. When another denominator is used, it is labeled explicitly.
- Displayed payouts and minimum units can vary. The table placard and confirmed house procedure control the actual game.
If you need the shorter introduction, begin with Craps Math. For a bet-by-bet lookup without the proofs, use the craps payout table.
Notation used in the workbook
| Symbol | Meaning | Example |
|---|---|---|
| P(A) | Probability that event A occurs | P(7) = 6/36 = 1/6 |
| W(n) | Number of ordered ways to roll total n | W(9) = 4 |
| B | Initial wager | B = $10 |
| E | Expected net result per wager or decision | E = −$0.14 |
| H | House edge under the stated denominator | H = |E|/B |
| g | Odds against an event | If P(A)=1/3, then g=2, or 2 to 1 against |
A negative expected value is an expected loss. For example, E = −$0.14 means an average loss of 14 cents per decision over a very large number of identical decisions. It does not mean every $10 bet loses 14 cents, and it does not constrain the result of one session.
The 36 ordered dice outcomes
Treat the dice as distinguishable—red and green, for example. A red 1 with a green 3 is a different ordered outcome from red 3 with green 1. That is why rolling 4 has three ways: 1–3, 2–2, and 3–1.
| Total | Ordered combinations | Ways | Probability |
|---|---|---|---|
| 2 | 1–1 | 1 | 1/36 |
| 3 | 1–2, 2–1 | 2 | 2/36 |
| 4 | 1–3, 2–2, 3–1 | 3 | 3/36 |
| 5 | 1–4, 2–3, 3–2, 4–1 | 4 | 4/36 |
| 6 | 1–5, 2–4, 3–3, 4–2, 5–1 | 5 | 5/36 |
| 7 | 1–6, 2–5, 3–4, 4–3, 5–2, 6–1 | 6 | 6/36 |
| 8 | 2–6, 3–5, 4–4, 5–3, 6–2 | 5 | 5/36 |
| 9 | 3–6, 4–5, 5–4, 6–3 | 4 | 4/36 |
| 10 | 4–6, 5–5, 6–4 | 3 | 3/36 |
| 11 | 5–6, 6–5 | 2 | 2/36 |
| 12 | 6–6 | 1 | 1/36 |
The mental shortcut is W(n)=n−1 for totals 2 through 7 and W(n)=13−n for totals 8 through 12. The sequence is therefore 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1.
Four probability rules used throughout
1. Equally likely outcomes
When all elementary outcomes are equally likely, P(A) = favorable outcomes ÷ total outcomes. Rolling 9 has four favorable ordered outcomes out of 36, so P(9)=4/36=1/9.
2. Sum rule
For events A and B, P(A or B)=P(A)+P(B)−P(A and B). One roll cannot be both 7 and 9, so those events are mutually exclusive and the overlap is zero: P(7 or 9)=6/36+4/36=10/36=5/18.
The six point totals have 3+4+5+5+4+3=24 combinations. A come-out roll therefore establishes a point with probability 24/36=2/3.
3. Product rule for independent rolls
Legal rolls are modeled as independent. For independent events, P(A and B)=P(A)×P(B). Two consecutive 7s therefore have probability (1/6)×(1/6)=1/36. The first 7 does not make the second more or less likely.
4. Conditional probability
P(A|B)=P(A and B)/P(B). Once a point is established, rolls that are neither the point nor 7 postpone the decision. We can condition on the next decisive roll. With point 9, four combinations win and six lose, so P(9 before 7)=4/(4+6)=2/5.
| Point | Ways to make point | Ways to make 7 | Pass wins | Odds against Pass |
|---|---|---|---|---|
| 4 or 10 | 3 | 6 | 3/9 = 1/3 | 2 to 1 |
| 5 or 9 | 4 | 6 | 4/10 = 2/5 | 3 to 2 |
| 6 or 8 | 5 | 6 | 5/11 | 6 to 5 |
Convert probability, true odds, and payout odds
If an event has probability p, the odds against it are g=(1−p)/p=1/p−1. If the odds against an event are a:b, its probability is b/(a+b). These are mathematical odds, not necessarily what a casino pays.
| Event | Probability | True odds against | Typical casino payout |
|---|---|---|---|
| Roll 11 next | 2/36 = 1/18 | 17 to 1 | 15 to 1 |
| Roll 12 next | 1/36 | 35 to 1 | 30 to 1 |
| Point 6 before 7 | 5/11 | 6 to 5 | Place 6 pays 7 to 6; Pass Odds pays 6 to 5 |
A payout can be written “to” or “for.” A 15-to-1 win returns the original wager plus 15 units of profit. A 16-for-1 return includes the original unit and is economically identical. Confusing those conventions creates one-unit payoff errors.
Expected value and house edge
For mutually exclusive results i, expected value is:
E = Σ [probability of result i × net profit or loss for result i]
For a simple wager that wins net W with probability p and loses wager B otherwise, E=pW−(1−p)B. Under the common initial-wager convention, H=−E/B when E is negative.
Denominator warning: “House edge,” “expected loss per resolved decision,” “edge on total money at risk,” and casino “hold” are not interchangeable. A percentage is incomplete until you know what is in its denominator.
Exact Pass Line probability and edge
Pass wins immediately on come-out 7 or 11: 8/36=2/9. If a point is established, multiply the chance of establishing that point by the conditional chance of repeating it before 7.
| Point pair | Chance one point is established | Chance Pass then wins | Winning contribution for one point |
|---|---|---|---|
| 4 and 10 | 3/36 each | 1/3 | (3/36)(1/3) = 1/36 each |
| 5 and 9 | 4/36 each | 2/5 | (4/36)(2/5) = 2/45 each |
| 6 and 8 | 5/36 each | 5/11 | (5/36)(5/11) = 25/396 each |
P(Pass win)=2/9+2(1/36+2/45+25/396)=244/495≈49.293%. Therefore P(Pass loss)=251/495≈50.707%.
At even money, a $1 flat bet has E=(244/495)(+$1)+(251/495)(−$1)=−7/495 dollars. The edge is 7/495≈1.414%.
Exact Don’t Pass probability: why 1.36% and 1.40% both appear
Under standard bar-12 rules, Don’t Pass wins immediately on 2 or 3, loses on 7 or 11, and pushes on 12. After a point, it wins when 7 arrives first.
| Result per initial wager | Exact probability | Approximate probability |
|---|---|---|
| Win | 949/1980 | 47.929% |
| Lose | 976/1980 = 244/495 | 49.293% |
| Push on 12 | 55/1980 = 1/36 | 2.778% |
The win fraction comes from 3/36 for immediate 2 or 3 plus the point paths:
P(win)=3/36+2[(3/36)(6/9)+(4/36)(6/10)+(5/36)(6/11)]=949/1980.
Including the push as an initial wager, a $1 flat bet has E=(949−976)/1980=−27/1980=−3/220. The conventional edge is therefore 3/220≈1.364%.
If you exclude pushes and condition only on resolved decisions, the denominator becomes 1925 rather than 1980. The win and loss probabilities are then 949/1925 and 976/1925, producing an edge of 27/1925≈1.403%. Nothing about the game changed; only the denominator did.
The geometric-series view of repeated pushes
A player can see several come-out 12s before a win or loss. The probability of “zero or more pushes, then event A” is P(A)[1+(1/36)+(1/36)²+…]. The bracket is a geometric series equal to 1/(1−1/36)=36/35. Multiplying 949/1980 by 36/35 gives 949/1925—the same resolved-decision win probability obtained by simply conditioning on “not 12.”
Why free Odds have zero edge but more risk
Pass/Come Odds pay the true point-versus-7 odds: 2 to 1 on 4/10, 3 to 2 on 5/9, and 6 to 5 on 6/8. For $10 Odds behind point 4, the bet wins $20 with probability 1/3 and loses $10 with probability 2/3:
E=(1/3)(+$20)+(2/3)(−$10)=0.
Adding a zero-edge component does not erase the expected loss on the flat bet. It increases total action while leaving the flat bet’s expected dollar loss unchanged. A $10 Pass Line decision still costs about 14.14 cents in expectation whether the player adds no Odds or the maximum; the larger combined wager can produce much larger session swings.
| Pass Odds rule | Average total action per initial $1 flat decision | Flat expected loss | Edge on average combined action |
|---|---|---|---|
| No Odds | $1.000 | 7/495 | 1.414% |
| 1× | $1.667 | 7/495 | 0.848% |
| 2× | $2.333 | 7/495 | 0.606% |
| Full double | $2.472 | 7/495 | 0.572% |
| 3× | $3.000 | 7/495 | 0.471% |
| 3-4-5× | $3.778 | 7/495 | 0.374% |
| 5× | $4.333 | 7/495 | 0.326% |
| 10× | $7.667 | 7/495 | 0.184% |
The table’s denominator is average money wagered across all initial Pass decisions, including decisions that end on the come-out before Odds can be placed. That is why it differs from a point-only comparison. “Full double” means 2× on 4/5/9/10 and 2.5× on 6/8 to produce a whole 3× flat win; confirm how the casino uses the term.
Don’t Pass/Don’t Come Odds reverse the payout: lay 2 to win 1 behind 4/10, 3 to win 2 behind 5/9, and 6 to win 5 behind 6/8. Those components also have zero mathematical edge. However, casinos describe dark-side limits by maximum lay, maximum win, or an Odds multiple, so a universal “5×” combined-edge table can be misleading. Calculate from the actual allowed lay and the denominator you intend to report.
Place bets: derive all three edges
A Place bet ignores every roll except the chosen number and 7. Use the conditional point race and the displayed payout.
| Place number | Win probability | Standard payout | EV per $1 | House edge |
|---|---|---|---|---|
| 4 or 10 | 3/9 = 1/3 | 9 to 5 | (1/3)(9/5)−(2/3) = −1/15 | 6.67% |
| 5 or 9 | 4/10 = 2/5 | 7 to 5 | (2/5)(7/5)−(3/5) = −1/25 | 4.00% |
| 6 or 8 | 5/11 | 7 to 6 | (5/11)(7/6)−(6/11) = −1/66 | 1.52% |
Example: a $6 Place 6 wins $7 with probability 5/11 and loses $6 with probability 6/11. E=(5/11)(+$7)+(6/11)(−$6)=−$1/11, which is −$0.0909 per resolved $6 wager. Dividing by $6 produces 1/66≈1.52%.
Buy bets: commission timing changes the answer
A Buy bet pays true odds but charges a commission. The old shortcut “all Buy bets have 4.76% edge” is only valid under a particular upfront-commission denominator and ignores win-only collection and chip rounding.
| $20 Buy 4 rule | Net if 4 wins | Net if 7 wins | Expected value | Reported edge |
|---|---|---|---|---|
| $1 commission paid upfront | +$39 | −$21 | (1/3)(39)+(2/3)(−21)=−$1 | 4.76% of $21 total outlay; 5% of $20 base bet |
| $1 commission collected only on a win | +$39 | −$20 | (1/3)(39)+(2/3)(−20)=−$1/3 | 1.67% of $20 base bet |
Real tables may round commission or restrict Buy amounts. On 4/10, win-only vig often makes Buy preferable to Place at practical thresholds; on other numbers the crossover depends on collection timing and rounding. Use the complete Buy bet comparison before applying a generic rule.
Lay and Place-to-Lose bets
A Lay bet wins when 7 appears before the selected number and pays true odds after commission. Because 7 is favored, the player risks more than the potential win.
| Lay number | Lay amount | True win | 5% commission on win | Upfront edge using total outlay |
|---|---|---|---|---|
| 4 or 10 | $40 | $20 | $1 | $1/$41 = 2.44% |
| 5 or 9 | $30 | $20 | $1 | $1/$31 = 3.23% |
| 6 or 8 | $24 | $20 | $1 | $1/$25 = 4.00% |
Those last-column figures assume the $1 is paid whether the wager wins or loses. If the commission is collected only on a win, the effective edges are lower. See the Lay bet guide for both conventions and rounding examples.
Some regulated rules also list Place-to-Lose payouts. These are not the same as true-odds Lay bets and do not add a separate commission:
| Place to lose | Win probability | Payout | House edge |
|---|---|---|---|
| Against 4 or 10 | 6/9 = 2/3 | 5 to 11 | 1/33 = 3.03% |
| Against 5 or 9 | 6/10 = 3/5 | 5 to 8 | 1/40 = 2.50% |
| Against 6 or 8 | 6/11 | 4 to 5 | 1/55 = 1.82% |
Field, Horn, C&E, and one-roll proposition math
For a one-roll wager, include every possible next-roll result and use net—not gross—payout. The table assumes common 30-to-1 payouts on 2/12 and 15 to 1 on 3/11 unless the row states otherwise.
| Wager | Winning combinations | Key net outcomes | House edge |
|---|---|---|---|
| Field, 2 and 12 both pay 2 to 1 | 16 win; 20 lose | 2/12 win +2; other Field wins +1; 5/6/7/8 lose −1 | 2/36 = 5.56% |
| Field, 12 pays 3 to 1 | 16 win; 20 lose | 12 wins +3 instead of +2 | 1/36 = 2.78% |
| Any Craps | 4 win; 32 lose | +7 or −1 | 1/9 = 11.11% |
| Eleven | 2 win; 34 lose | +15 or −1 | 1/9 = 11.11% |
| 2 or 12 straight up | 1 win; 35 lose | +30 or −1 | 5/36 = 13.89% |
| Any 7 | 6 win; 30 lose | +4 or −1 | 1/6 = 16.67% |
| C&E | 4 craps, 2 eleven, 30 lose | Split equally: net +3 on craps component or +7 on eleven component before the other half loses | 1/9 = 11.11% |
Horn derivation
A four-unit Horn places one unit each on 2, 3, 11, and 12. A 2 or 12 nets 30 winning units minus three losing units, or +27. A 3 or 11 nets +12. All other 30 combinations lose four units.
E=[2(1/36)(27)+2(2/36)(12)+(30/36)(−4)] units=−1/2 unit. Divide the half-unit loss by the four-unit total wager: Horn edge = 1/8 = 12.50%.
World/Whirl derivation
A five-unit World is a four-unit Horn plus one unit on Any 7. With the stated payouts, 2/12 net +26 units, 3/11 net +11, 7 pushes the combined wager, and the other 24 combinations lose five units. The expected loss is 2/3 of one unit per five units wagered, producing 13.33%.
Hardway math
A Hardway is not a one-roll bet. Irrelevant totals postpone the decision. Condition on the hard combination versus the easy combinations and 7.
| Hardway | Win ways | Lose ways | Win probability | Payout | House edge |
|---|---|---|---|---|---|
| Hard 4 or 10 | 1 hard combination | 2 easy + 6 seven = 8 | 1/9 | 7 to 1 | 1/9 = 11.11% |
| Hard 6 or 8 | 1 hard combination | 4 easy + 6 seven = 10 | 1/11 | 9 to 1 | 1/11 = 9.09% |
For Hard 6, for example, E=(1/11)(+$9)+(10/11)(−$1)=−1/11. The full Hardway guide covers working status and table procedure.
Big 6 and Big 8 proof
Big 6/8 pays even money on the same five-versus-six race used by Place 6/8. For a $1 unit, E=(5/11)(+$1)+(6/11)(−$1)=−1/11, so the edge is 9.09%. Place 6/8 pays 7 to 6 and has only a 1.52% edge under standard rules. The underlying dice race is identical; the worse payout creates the difference.
Verified exercises and answers
1. What is the chance a come-out roll establishes a point?
The point totals contain 24 combinations, so 24/36=2/3.
2. After a point, how many additional rolls are expected?
When point n has W(n) ways, the decision probability per roll is [W(n)+6]/36. A geometric waiting time has mean 36/[W(n)+6]: four rolls for 4/10, 3.6 for 5/9, and 36/11≈3.273 for 6/8.
3. How many total rolls does an average Pass decision take?
Start with the come-out roll and weight each point’s additional waiting time by its come-out probability: 1+2[(3/36)(4)+(4/36)(3.6)+(5/36)(36/11)]=557/165≈3.376 rolls.
4. Can a Pass Line bet be placed after a point?
House procedure varies. When accepted, it is effectively a Put bet. A flat late bet skips the favorable come-out: its point-only edge at even money is 33.33% on 4/10, 20% on 5/9, and 9.09% on 6/8. Permitted true Odds can improve the blended price enough to compete with a Place bet, so “never” is too broad; calculate the actual construction.
5. Can Don’t Pass be removed after a point?
Many regulated rules permit reducing or removing an established Don’t Pass bet but prohibit replacing or increasing the removed amount. From the established point onward, the Don’t side has the favorable six-versus-point-number race, so removing it gives up that conditional advantage. That observation does not erase the come-out risk already taken and does not turn the complete wager positive expectation.
6. What fair even-decision payout would Pass require?
Pass wins 244/495 and loses 251/495. A fair net payout k solves k(244/495)−251/495=0, so k=251/244≈1.0287 to 1. The casino pays 1 to 1, creating the 1.414% edge.
7. How often does a winning Pass bet win immediately?
P(natural | Pass win)=(8/36)/(244/495)=55/122≈45.08%. This is a conditional statement about winning decisions, not the probability that the next come-out is a natural.
How to audit any craps wager yourself
- Write the exact rule, including when the bet works, pushes, commission timing, and whether the displayed payout is “to” or “for.”
- List every mutually exclusive result that resolves the wager.
- Count ordered dice combinations for each result.
- For a multi-roll wager, condition on only the outcomes that decide it.
- Calculate net profit after losing components and commission—not gross chips returned.
- Multiply each net result by its probability and add the products.
- State the denominator before converting expected loss into a percentage.
- Test the formula with a clean wager amount that avoids rounding.
Sources and calculation policy
- Massachusetts Gaming Commission: Craps and Mini-Craps rules for permitted wagers, standard minimum payouts, Odds, removal rules, and dice procedure.
- OpenStax Introductory Statistics: expected value and standard deviation for the general expected-value framework.
- Every combination count, conditional probability, exact fraction, expected-value result, and rounded percentage on this page was independently recomputed from the stated assumptions during the August 18, 2026 review.
Responsible-play note: Lower edge means lower expected cost per dollar wagered, not safety or guaranteed profit. More Odds, more simultaneous bets, and longer play can all increase the dollars exposed. Set a loss limit before gambling and never chase a result predicted by a formula.