Two dice, 36 outcomes
Chapter 1: Foundations · Lesson 1 of 21 · 6 min read
Count the 36 ways two dice can land and see why 7 rules the table.
Craps is played with two ordinary six-sided dice. Each die is fair, so every face has the same chance.
Paint one die red and one blue. The red die can land 6 ways and the blue die 6 ways, which makes 6 × 6 = 36 combinations, each just as likely as the others. A red 1 with a blue 6 is a different combination from a red 6 with a blue 1, even though both total 7.
What matters in craps is the total. Totals are not equally likely, because some have more combinations:
| Total | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Ways | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |
| Chance | 2.8% | 5.6% | 8.3% | 11.1% | 13.9% | 16.7% | 13.9% | 11.1% | 8.3% | 5.6% | 2.8% |
Notice the shape: the ways climb by one up to 7, then fall again. 7 is the most likely total, at 6 ways out of 36, or 1 in 6. Nearly every bet in craps is a bet on whether some number shows up before a 7 does, so it pays to remember the column of "ways".
A trick for remembering: totals the same distance from 7 have the same number of ways. 6 and 8 both have 5 ways, 5 and 9 have 4, 4 and 10 have 3, 3 and 11 have 2, and 2 and 12 have 1.
Some totals can be rolled as a pair (both dice the same): 2-2 is a "hard 4", 3-3 a "hard 6", 4-4 a "hard 8" and 5-5 a "hard 10". Any other way to make those totals is "easy". Hard 6 is one combination; easy 6 is four (1-5, 5-1, 2-4, 4-2).
Show the math: Counting the 36 combinations
Each die shows 1 to 6 independently, so the outcomes are the ordered pairs (red, blue): 6 × 6 = 36, each with probability 1/36.
Ways to make total t: for t from 2 to 7 it is t − 1; for t from 7 to 12 it is 13 − t. Check: 1+2+3+4+5+6+5+4+3+2+1 = 36.
P(7) = 6/36 = 1/6 ≈ 16.67%. P(6) = 5/36 ≈ 13.89%.
Try it yourself. No betting yet: just roll and watch the totals.
Roll the dice
Interactive practice table. Try this in the app (free, play money).
Now let the computer do the rolling. Over a few hundred rolls the shape from the table appears on its own.
7 vs 2, 600 times
Interactive practice table. Try this in the app (free, play money).
The tally chart is a histogram of the 11 totals, with the expected counts drawn as faint outlines.
How many equally likely combinations do two dice have?
- 11
- 12
- 36
- 21
Show answer
C: 36. 6 faces × 6 faces = 36. The 11 totals are not equally likely.
Which total is most likely?
- 6
- 7
- 8
- They're all equal
Show answer
B: 7. 7 can be made 6 ways (1-6, 2-5, 3-4, 4-3, 5-2, 6-1), more than any other total.
How many ways can you roll a 9?
- 3
- 4
- 5
- 6
Show answer
B: 4. 3-6, 4-5, 5-4, 6-3. Shortcut: 9 is two away from 7, so it has 6 − 2 = 4 ways.
Which of these is a "hard 8"?
- 2-6
- 3-5
- 4-4
- 6-2
Show answer
C: 4-4. Hard means a pair. 4-4 is the only hard 8; the other four ways are easy 8s.