Sic Bo Probability Calculator

Leave Dice Total at 10 and Stake at $10. Calculate. Probability is 12.50% and Fair Payout on Stake is $70.00. Three six-sided dice have 6³ = 216 ordered triples. Exactly 27 of those triples sum to 10, so 27/216 = 1/8 = 12.50%. A fair book would pay 7-to-1 profit on a winning $10 ticket, which is $70 profit and $80 returned. The field labelled Fair Payout on Stake is that $70, computed as stake × (216/ways − 1). Casinos do not pay that price. A typical specific-total 10 or 11 pays 150-to-1 or 180-to-1 on triples of a kind only, and a much thinner 5-to-1 or 6-to-1 on the total itself, depending on the layout. Compare the felt to $70; the gap is the house edge.

Change Dice Total to 3. Probability falls to 0.46% (one way: 1-1-1) and fair payout jumps to $2,150.00, which is 215-to-1. Layouts that pay 150-to-1 or 180-to-1 on a triple one are taking a large slice of a rare event. Total 18 (6-6-6) is the mirror image: same 1 way, same 0.46%, same $2,150 fair profit on $10. Totals 4 and 17 have 3 ways, 1.39%, fair $710. Totals 5 and 16 have 6 ways, 2.78%, fair $350. The ways array the engine uses is [1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1] for totals 3 through 18.

Stake scales the dollar output only. A $25 stake on total 10 still shows 12.50% and a fair profit of $175. Probability does not care about chip size. Use Stake to translate the fair multiplier into the currency you actually put down, then subtract whatever the casino prints on the cloth.

This is not a small-big-odd-even calculator. Those even-money-ish bets have their own way counts (small is totals 4–10 excluding triples, 105 ways, and so on). Type a total from 3 to 18 inclusive. Anything outside that range has zero ways. Do not type 9.5; totals are integers.

After you have the fair number, look at the actual posted payout for that specific total. If the layout pays 6-to-1 on a 10, your $10 win is $60, not $70, and the edge is 1 − 0.125 × 7 = 12.5% on a to-1 reading, or compute EV = 0.125 × 60 + 0.875 × (−10) = −2.50 on a $10 stake, a 25% hold if they pay 6-for-1 including stake. Read the felt's "to" versus "for" language before you call the gap.

Sic bo is a three-dice total game that travelled from China to Southeast Asian casinos and then to the Strip as a carnival layout with a dozen proposition boxes. The Chinese name 骰寶 (dice treasure) and the French-colonial "sic bo" both describe the same 216-cell sample space. Macau still books more sic bo than craps; North American rooms treat it as a high-edge side attraction. This calculator prices one slice of that layout: a bet that the three-dice sum equals a chosen total from 3 through 18.

The fair multiplier is 216 divided by the number of ordered triples that realise that sum. Casinos never pay the fair multiplier on the interesting boxes. Triple 1 and triple 6 might post 150:1 or 180:1 against a fair 215:1. A total of 4 or 17 might post 50:1 or 60:1 against a fair 71:1. A total of 9, 10, 11, or 12 — the mode of the distribution — is paid at a short price that looks generous next to even money and is not. The tool's job is to put the fair number next to your stake so the posted number has a denominator.

History of the way counts is just generating functions. The number of ways to score s with three dice is the coefficient of x^s in (x + x² + x³ + x⁴ + x⁵ + x⁶)³, which expands to the palindromic sequence 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1. That sequence is older than any casino. It is the same sequence a backgammon player uses for a two-dice pip count, extended by one die. Nothing about a "hot" table changes the 27 ways to make 10.

Related bets on the same cloth — small, big, odd, even, specific triples, any triple, two-dice combinations, and the 4-5-6 / 1-2-3 instant decisions in some Asian variants — each have their own way counts and posted prices. Do not feed those into Dice Total. A "small" wager is not a total of 9. Mixing them is how players convince themselves a 12% edge is a 2% edge.

Limits: the calculator assumes three fair distinguishable dice (ordered triples). A physically biased die, a cage that does not tumble, or an electronic sic bo with a published but non-uniform RNG would invalidate the 216-cell model. It also assumes you are pricing the total itself, not a "total and a triple" parlay. Fair Payout is the profit a zero-edge book would pay, not a prediction of the next spin of the cage.

Three independent fair dice produce 216 equally likely ordered triples. Galileo's note on three dice, Sopra le scoperte dei dadi, already counted the ways for each total and showed that ten and eleven are equally frequent, contrary to gamblers who thought nine and ten should match. The way counts from total 3 through total 18 are 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1. Those sixteen numbers sum to 216. Probability of a named total is ways over 216. Fair multiplier, including stake, is 216 divided by ways. Totals 10 and 11 each have 27 ways, so 27/216 equals 12.50 percent, and the fair multiplier is 216/27 equal to 8 for one.

Enumeration check for ten, which is the worked default. Partitions of 10 into three parts each between 1 and 6: (6,3,1) and its six permutations, (6,2,2) and its three, (5,4,1) and its six, (5,3,2) and its six, (4,4,2) and its three, (4,3,3) and its three. Six plus three plus six plus six plus three plus three equals 27. The same coefficient falls out of the generating function (x plus x squared plus … plus x to the sixth) cubed, the x-to-the-tenth term. Mirror identity: ways for s equal ways for 21 minus s, because each face x maps to 7 minus x. That is why 10 and 11 both have 27, 9 and 12 both have 25, and the fair payouts are palindromes.

Worked default total 10, stake 10. Probability 12.50 percent. Fair profit on a win, stake returned plus seven units of profit if the book paid the fair 7-to-1, would be 70 on a 10 stake, and expected value would be 0.125 times 70 plus 0.875 times minus 10, equal to zero. House edge against a posted to-one price r is 1 minus P times (r plus 1), or compute expected value as P times r B plus (1 minus P) times minus B, then divide by B. A layout paying 6-to-1 on a 10: expected value over stake equals 0.125 times 6 minus 0.875, equal to minus 0.125, a 12.5 percent edge. Five-to-one: minus 0.250, a 25 percent edge. The page reports the fair profit so that difference can be formed against whatever the cloth prints.

Assumptions. Ordered triples, not unordered colour-indistinguishable dice; the 216 count already treats (6,3,1) as distinct from (1,3,6). Uniform faces. No memory between throws. Totals outside 3 through 18 have zero ways. Small and big bets, triples, and specific-doublets are different generating functions and are not this total-bet lecture. The cage has no memory. Two hundred sixteen is the whole sample space.

History sits in the generating function as much as in Galileo. The binomial-looking way list 1, 3, 6, 10, 15, 21, 25, 27 is the truncated convolution of three uniform six-sided variables, peaking at 10 and 11 rather than at the naive 3 times 3.5. Anyone quoting 10 as "more due" after a run of 9s is quoting the wrong model. The 27 ways do not accumulate from previous throws. They reset every shake.