Three dice instead of two
Sic bo is a game of chance with a built-in house edge, and no strategy, system or timing changes that in the player’s favor over time. Most players lose money over time, and nobody should gamble money they cannot afford to lose. The game is a good way to learn how counting works when a third die is added.
In sic bo, three dice are shaken in a covered container or cage, and the result is revealed. Players place chips on a layout that shows many different bets, resolved by the three faces shown. The game has roots in traditional dice games, part of a long dice history, and is common in some casinos. Layouts and payouts vary by venue.
216 outcomes
Each die has six faces, so three dice have 6 × 6 × 6 = 216 equally likely ordered outcomes. Every bet in the game is priced against this count.
The totals from 3 to 18 are not equally likely. The number of ways to make each total is:
| Total | Ways | Total | Ways |
|---|---|---|---|
| 3 | 1 | 18 | 1 |
| 4 | 3 | 17 | 3 |
| 5 | 6 | 16 | 6 |
| 6 | 10 | 15 | 10 |
| 7 | 15 | 14 | 15 |
| 8 | 21 | 13 | 21 |
| 9 | 25 | 12 | 25 |
| 10 | 27 | 11 | 27 |
Adding one column gives 108 and the pair of columns 216, as expected. The middle totals are far more common than the ends, for the same reason that 7 is common with two dice.
The main bets
Typical bets are listed below. Payouts are common but not universal.
Big and small. Small wins on totals of 4 to 10, and big on 11 to 17. A triple (all three dice alike) loses both. Of the 107 outcomes with totals from 4 to 10, two are triples (2-2-2 and 3-3-3), leaving 105 winning outcomes out of 216. The bet pays 1 to 1. Expected result per $1: (105 − 111) ÷ 216 = −6 ÷ 216, about −2.78%. Big is the mirror image and costs the same. This is one of the cheapest bets, though it still carries an edge, and it is a good illustration of how even-money bets are not even.
Single number. The player picks a number from 1 to 6. It pays 1 to 1 if it appears on one die, 2 to 1 on two dice, and 3 to 1 on all three. The count: one die shows it in 75 outcomes, two dice in 15, three in 1, and none in 125. Expected result per $1: (75 × 1 + 15 × 2 + 1 × 3 − 125) ÷ 216 = (108 − 125) ÷ 216 = −17 ÷ 216, about −7.87%.
Specific triple. One chosen triple, such as three 4s, can appear in exactly 1 of 216 outcomes. If it pays 150 to 1, the expected result per $1 is (150 − 215) ÷ 216 = −65 ÷ 216, about −30.1%.
Any triple. Six of the 216 outcomes are triples. At 24 to 1, the expected result is (6 × 24 − 210) ÷ 216 = −66 ÷ 216, about −30.6%.
A total. Betting on a total of 4 or 17 wins in 3 of 216 outcomes. At 60 to 1, the result is (3 × 60 − 213) ÷ 216 = −33 ÷ 216, about −15.3%. A total of 10 has 27 ways, and at 6 to 1 the result is (27 × 6 − 189) ÷ 216 = −27 ÷ 216, which is −12.5%.
All figures use the stated payouts, which differ between venues. Always check the posted payout.
Why triples and totals pay as they do
The payout for each bet scales with its probability: rarer outcomes pay more. A specific triple is a 1-in-216 event, so the headline payout is large. The edge arises because the payout is always set below the true odds. In this game the gap grows for the rarest events. The long-shot bets with the biggest payouts carry some of the biggest edges, a pattern seen at craps with proposition bets.
Triples are also the reason big and small cost something. Because triples make both lose, the casino wins on 6 of 216 outcomes automatically, which is where the small edge on those bets comes from.
Other three-dice games
Several other games use three dice. A carnival-style game often called chuck-a-luck or birdcage lets the player bet on a number and pays 1, 2 or 3 to 1 on the number of matching dice, which is the same single-number structure described above and has the same math. Many dice games and gambling traditions use three dice because the outcomes, though more numerous than with two dice, can still be counted by hand.
What the counts do and do not tell you
Counting gives a precise description of each bet’s average cost. It does not predict any individual throw. The dice are independent events, so a long stretch without a triple does not make one more likely. The tidy table of totals describes long-run frequencies; it is the law of large numbers at work, not a sequence to follow.
Anyone who chooses to play should treat their stake as the cost of entertainment, set a limit of spare money in advance, and stop when it is reached. A local support service can help anyone concerned about their gambling.