# The Basic Math of Craps: Required Understanding for Smart Players

As you know, craps is played with two dice.  (By the way, the little dots on a die are called “pips.”  A pip is simply a dot that symbolizes numerical value.  You find pips on things like dice and dominoes.)  Thirty-six (36) combinations of numbers can be rolled with two dice.  Refer to the table below.

Combinations Using Two Dice

DIE #1DIE #22-DICE TotalDIE #1DIE #22-DICE Total
112415
123426
134437
145448
156459
1674610
213516
224527
235538
246549
2575510
2685611
314617
325628
336639
3476410
3586511
3696612

As you can see in the table above, the possible total values when using two dice range from 2 to 12.  It’s important to know, without even thinking about it, the number of ways to roll each number 2 through 12.  This basic information is used as the basis for determining many aspects of the game, such odds and house advantage.

Ways to Roll a Number With Two Dice

2-DICE VALUE# OF WAYS TO ROLL THE 2-DICE VALUE
21
3 2
4 3
5 4
6 5
7 6
8 5
9 4
10 3
11 2
121

The table below is a common format for displaying the number of ways to roll each number, and the possible 2-dice combinations for rolling each number.

Ways to Roll a Number and Their Combinations

2-DICE VALUE2-DICE COMBINATIONS FOR ROLLING THE NUMBER# OF WAYS TO ROLL THE NUMBER
2[1-1]1
3[1-2] [2-1]2
4[2-2] [1-3] [3-1]3
5[1-4] [4-1] [2-3] [3-2]4
6[3-3] [1-5] [5-1] [2-4] [4-2]5
7[1-6] [6-1] [2-5] [5-2] [3-4] [4-3]6
8[4-4] [2-6] [6-2] [3-5] [5-3]5
9[3-6] [6-3] [4-5] [5-4]4
10[5-5] [4-6] [6-4]3
11[5-6] [6-5]2
12[6-6]1

To ensure we’re reading the table above correctly, let’s look at an example.  Suppose we want to know how many ways there are to roll an 8 with two dice, and we want to know what two-dice combinations can result in an 8.  We see from the table above that there are five (5) ways to roll an 8, and the two-dice combinations are:

• 4 on die #1, and 4 on die #2.
• 2 on die #1, and 6 on die #2.
• 6 on die #1, and 2 on die #2.
• 3 on die #1, and 5 on die #2.
• 5 on die #1, and 3 on die #2.

Memorize the summary table above.  Without having to think about it, know how many ways there are to roll each number 2 through 12.

There’s an easy trick to help memorize it.

As shown in the table above, there are six ways to roll a 7, five ways to roll a 6 or 8, 4 ways to roll 5 or 9, three ways to roll a 4 or 10, two ways to roll a 3 or 11, and one way to roll a 2 or 12.  Notice that all the numbers, except for the number 7, are paired based on how many ways to roll them.  Let’s construct a table to help us memorize the pairings.  We’ll build one column at a time so it’s easy to follow.  Let’s start by putting the pairings in the first column, which are based on the number of ways to roll each number in the pairing.

Ways to Roll a Number by Pairings

2-Dice Pairings
7
6 or 8
5 or 9
4 or 10
3 or 11
2 or 12

Now, notice that each pairing has a number lower than 7 and a number higher than 7.  The first pairing is 7 minus one (i.e., 6) and 7 plus one (i.e., 8).  So, the first pairing is “6 or 8.”  The second pairing is 7 minus two (i.e., 5) and 7 plus 2 (i.e., 9).  So, the second pairing is “5 or 9.”  Do this technique for the remaining pairings.  To help you memorize the pairings, just think, “One away from 7 in both directions is 6 and 8.  Two away from 7 in both directions is 5 and 9.  Three away from 7 in both directions is 4 and 10.  Four away from 7 in both directions is 3 and 11.  And five away from 7 in both directions is 2 and 12.

Now, we need to figure out how many ways to roll each number.  The little math trick continues.  All you have to do is subtract one (1) from the low number of each pairing.  Let’s fill in the middle column of the table.  Remember, take the low number from each paring and subtract one, as shown in the middle column below.  Although the number 7 stands alone, you still subtract one from it.

Ways to Roll a Number by Pairings

PairingsLow # of the Pairing - 1
77 - 1 = 6
6 or 86 - 1 = 5
5 or 95 - 1 = 4
4 or 104 - 1 = 3
3 or 113 - 1 = 2
2 or 122 - 1 = 1

Ways to Roll a Number by Pairings

PairingsLow # of the Pairing - 1Ways to Roll the Number in Each Pairing
77 - 1 = 66
6 or 86 - 1 = 55
5 or 95 - 1 = 44
4 or 104 - 1 = 33
3 or 113 - 1 = 22
2 or 122 - 1 = 11

Let’s see if you’re paying attention.  Without looking at the table, how many ways are there to roll a 4?  If you can’t memorize it, then do the little math trick.  The pairing is “4 or 10,” and 4 is the low number of the pairing, so 4 – 1 = 3.  Therefore, there are three ways to roll a 4.

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The basic data in the table above is very important, so memorize it.  If you need help, memorize the pairings and do the simple math trick described above to figure it out.  If you want to win at craps, then you must know and understand this basic relationship among the numbers.

You may be thinking, “Why is this so important?  I hate math.  I use a calculator to add 2 + 2.  Can’t I just play the game without knowing all this gobbledygook?”  No!  It’s easy to see that craps is a game of odds where possible winning two-dice combinations are compared to possible losing two-dice combinations.  Whether numbers are winners or losers depends entirely the type of bets that you make, which we’ll discuss in other articles.  A number may be a winner when making one type of bet, while the same number may be a loser when making a different bet.  Regardless, the game boils down to understanding the relationship among the numbers, particularly how the number 7 relates to all the other numbers.  If you don’t understand it, you’re going to make stupid mistakes and you’re going to lose.  If you want to be a rock-solid player and win consistently, then you must memorize it.

For example, suppose we want to compare the number 7 to the number 10.  Suppose we want to bet on the 10 appearing on a roll before a 7 appears.  Therefore, our bet wins if a 10 shows before a 7, and our bet loses if a 7 shows before a 10.  Assume that all other numbers don’t matter, so we ignore them and keep rolling until either a 10 shows (we win) or a 7 shows (we lose).  Let’s bet \$1 and assume it’s an even‑money bet, which means if we lose, we lose the \$1, and if we win, we win \$1.  The odds for this even‑money bet are expressed as 1:1 (stated as “one to one”).  An even-money bet, or a 1:1 bet, means for each unit we bet and win, we win that exact amount.  For example, if we bet \$5 and win, then we win \$5; or if we bet \$8 and win, then we win \$8.  Is betting the number 10 against the 7 for even money a good bet?  No way!  It’s a terrible bet because we have twice as many chances of losing than winning.

From the table above, we see there are six ways to roll a 7, and only three ways to roll a 10.  That means there are twice as many ways for us to lose as there are for us to win.  So, making this even-money bet is not only terrible, it’s stupid.

To illustrate this further, let’s roll the dice 36 times and assume the results are distributed exactly according to the number of ways to roll each number (i.e., a perfect distribution).  After 36 rolls with a perfect distribution, we would see the following results:

• On 6 of the rolls, a 7 will show.
• On 5 of the rolls, a 6 will show.
• On 5 of the rolls, an 8 will show.
• On 4 of the rolls, a 5 will show.
• On 4 of the rolls, a 9 will show.
• On 3 of the rolls, a 4 will show.
• On 3 of the rolls, a 10 will show.
• On 2 of the rolls, a 3 will show.
• On 2 of the rolls, an 11 will show.
• On 1 roll, a 2 will appear.
• And on one roll, a 12 will appear.

Note: The above outcome over 36 rolls occurs only with a perfect distribution, which is unlikely, but useful for illustrating how and why odds are important in the game of craps.

Let’s use the same example except this time we get the 2:1 odds that we expect to get when we bet the 10 against the 7 (i.e., instead of getting only 1:1 even money).  If we again bet \$1 on each of the 36 rolls, we expect to win \$2 three times (when the 10 shows) and lose \$1 six times (when the 7 shows).  Therefore, for a 2:1 odds bet, our net result is that we break even, as we expect (i.e., we win \$2 three times, or \$2 x 3 = \$6; and we lose \$1 six times, or \$1 x 6 = \$6).  So, with the 2:1 odds bets, we win \$6 and we lose \$6 (i.e., we break even as we expect).

Since 10 is our favorite number, let’s look at Place betting the 10 against the 7.  As we know from the table above, there are three ways to roll a 10.  If we Place bet \$5 on the 10 against the 7, we expect to win \$10 (remember, there are six ways to roll a 7 and 3 ways to roll a 10, so we expect to get 2:1 odds on the 10, so we expect to get \$5 x 2 = \$10 when we win a \$5 bet).  Ready to find out how the casino screws us?

Remember, there are six ways to roll a 7, and 3 ways to roll a 10.  In a perfect distribution over 36 rolls, we expect a 7 to appear six times, and a 10 to appear three times.  On the six rolls when a 7 appears, we lose our \$5 bet.  On the three rolls when a 10 appears, we win the bet.  Because there are twice as many ways to roll a 7 as there are to roll a 10 (i.e., the odds are expressed as 2:1), we expect to get two times our \$5 bet when we win, or \$10.  So, for all three winning bets, we expect to win a total of \$30 (i.e., with a \$5 bet at 2:1 odds, and with the 10 appearing three times in 36 rolls in a perfect distribution, we win \$5 (our bet amount) x 2 (for the 2:1 odds) x 3 (for the three times the 10 appears in 36 rolls) = \$30.  Now, let’s look at how much we win when the casino only gives us 9:5 odds instead of the full 10:5 odds.  In this case, for all three winning bets, we expect to win a total of \$27 (i.e., with a \$5 bet at 9:5 odds, and with the 10 appearing three times in 36 rolls in a perfect distribution, we win \$5 (our bet amount) x 9 (for the 9:5 odds) x 3 (for the three times the 10 appears in 36 rolls) = \$27.  If the casino paid true odds, then over time, everything would even out and no one would make a profit.  However, the casino is in business to make money, so they only give us 9:5 odds (instead of 10:5), so over time when everything evens out, they end up making a \$3 profit from us.

The house advantage varies among the many different types of possible craps bets.  We’ll discuss them all later and you’ll learn which bets have high house advantages and which have relatively small house advantages.  Obviously, you want to avoid the bets with the higher house advantages and focus on those with the smallest.

I know what you’re thinking.  “If the casino has a built-in advantage on every bet which means I’m supposed to lose over time, why should I bother playing at all?”  Great question!  We play because there’s an important thing called “variance” that allows us to win.  We’ll go into detail about variance in another article, but variance is our friend at the craps table传奇sf.  Variance means you likely won’t see a perfect distribution of the results when rolling the dice.  Instead, distribution variance is what gives us the hot and cold streaks that are so common.  Knowing how to recognize those variances (i.e., hot and cold streaks) and knowing how to adapt your play to them are what enables us to win.  And winning is what it’s all about.  Don’t be a stupid player.  Be a rock-solid, smart player.  Make the casino fear you, hate you.  It’s fun asking the pit boss for a wheelbarrow to carry all your chips to the cage.

Here we list some more Craps Math and Statistics related posts:

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Written by John Nelsen in partnership with the team of craps pros at crapspit.org.