The 36 combinations: the math behind every craps payout

Every payout on a craps table is a price on one of 36 equally likely dice outcomes. If you can count to 36, you can check any number the casino quotes you. Here is the whole grid, what it says about each point, and the pass line edge worked out by hand.

The grid

Two dice, six faces each, land in 6 × 6 = 36 ways. Each of those 36 is exactly as likely as any other. A 3 on the first die and a 4 on the second is a different outcome from a 4 and a 3, even though both total 7. That distinction is the whole reason 7 is common and 2 is rare.

Read the diagonal from bottom-left to top-right and you'll see six 7s. Count the 2s and you'll find one. Tally every total and you get the list that every serious craps player has memorized:

The shape is a pyramid with 7 at the peak. Each step away from 7 loses one way. That symmetry is why the 4 and 10 are priced the same, the 5 and 9 are priced the same, and the 6 and 8 are priced the same on every bet on the layout.

Why the 7 dominates

Six ways out of 36 means a 7 shows on one roll in six. Nothing else comes close: the 6 and 8 each have five ways, and the 2 and 12 have one. Over a hundred rolls you'd expect about seventeen 7s, fourteen 6s, and three 12s.

That's why the game is built around the 7. On the comeout it's the number that wins for the pass line, because a bet that pays on the most common total is a good deal for the player. After a point is set, the 7 flips to the enemy, because now the most common total is the one that kills every bet on the table. The house builds its edge by letting you have the 7 when it's rare to need it and taking it back when it matters most. More on how those two phases interact in the comeout rules.

Point before seven: the only contest that matters

Once a point is set, every other total is noise. A roll of 5 when the point is 8 changes nothing. The pass line bet is now a race between two numbers: the point and the 7. So you can throw away the other 36 outcomes and just compare the ways.

Take the 4. Three ways make a 4 and six ways make a 7, so of the nine rolls that decide the bet, three go your way. You win one time in three, and lose two. A fair price for a bet you lose twice as often as you win is 2 to 1: lose $10 twice, win $20 once, break even. That's exactly what the free odds pay. For the 5, it's four ways against six, so you lose 6 for every 4 you win, and a fair payout is 6 to 4, which reduces to 3 to 2. For the 6, five ways against six gives 6 to 5.

Every other bet on a number is one of these ratios with a haircut. A place 4 pays 9 to 5 instead of 10 to 5. A place 6 pays 7 to 6 instead of 7.2 to 6. The haircut is the house edge, and you can now compute it yourself for any bet in the full ranking.

Deriving 1.41% for the pass line, by hand

The pass line resolves in two stages, so its edge is a sum of paths. Work in 36ths for the comeout and use the point-before-seven fractions above for the second stage.

Comeout wins. A 7 (6 ways) or 11 (2 ways) wins immediately: 8 of 36.

Comeout losses. A 2 (1), 3 (2), or 12 (1) loses immediately: 4 of 36.

Points. The other 24 of 36 rolls set a point, and each point wins with its own probability:

• 4 or 10: 3/36 to set it, then 3/9 to make it. Each contributes 3/36 × 3/9 = 1/36. Two of them: 2/36.
• 5 or 9: 4/36 to set it, then 4/10 to make it. Each contributes 16/360. Two of them: 32/360, which is 8/90.
• 6 or 8: 5/36 to set it, then 5/11 to make it. Each contributes 25/396. Two of them: 50/396.

Add the win paths: 8/36 + 2/36 + 8/90 + 50/396. Put everything over a common denominator of 1980 (that's 36 × 55): 440/1980 + 110/1980 + 176/1980 + 250/1980 = 976/1980. Divide top and bottom by 4 and you get the famous fraction: the pass line wins 244 out of 495 hands, or 49.29%. It loses the other 251 of 495.

The bet pays even money, so your expected result per $1 wagered is (244 − 251) / 495 = −7/495 = −0.01414. That is the 1.41% house edge, derived with nothing but counting. Per $100 on the pass line, the casino keeps $1.41 on average. The don't pass runs the same sum in reverse, with the 12 pushing instead of winning, and lands at 1.36%. Both are worked in dollars in the pass line article.

How long a hand lasts

A hand is one shooter's turn: it starts with a comeout and ends when they seven out. On average a hand runs about 8.5 rolls, though the average hides a wide spread. Plenty of hands end on the third or fourth roll, and hands of thirty rolls happen often enough that every regular has seen one. The math is the same either way: each roll is a fresh 36-way draw and the 7 shows one time in six no matter what came before.

Once a point is set, the expected number of rolls until the bet resolves depends on the point. A 4 or 10 has 9 deciding combinations out of 36, so on average it takes 4 rolls to resolve. A 6 or 8 has 11, so about 3.3 rolls. That's the fact behind the per-roll versus per-resolution caveat: a place 6 sits through several rolls, so its 1.52% per resolution spreads over more time than a field bet's 2.78% per roll does.

The gambler's fallacy, with dice

The gambler's fallacy is the belief that a number is “due” because it hasn't shown for a while, or that a table is “hot” because it has. It feels right. It's wrong, and the grid shows why.

Suppose the shooter has thrown fifteen rolls without a 7. That's unusual: the chance is (5/6) to the fifteenth power, about 6.5%. But the dice don't remember it. On the sixteenth roll, six of the 36 combinations still total 7 and thirty still don't. The chance of a 7 is one in six, the same as on the first roll and the same as it will be on the fiftieth. There's no debt to pay back, and no momentum to ride. A 7 that is “overdue” is exactly as likely as one that isn't.

The same logic kills every streak-based system. Pressing because the 6 has hit three times doesn't change the five ways to make a 6. Betting the don't because the table has been hot doesn't change the six ways to make a 7. What you can control is which bets you make and how much you risk, which is what bankroll management covers. The dice on the free table are audited publicly on the dice audit page, where a chi-square test checks that every one of the 36 combinations shows up as often as it should. That's what fair dice look like: no memory, no streaks that mean anything, and 7 one time in six forever.