## Craps Probability

Craps is played with two cube-shaped dice, each of which has six sides numbered with pips from one to six. When they come to rest after a roll, the total number of pips showing on the upward faces of both dice must be between 2 and 12, a pair of ones (snake eyes) being the lowest and a pair of sixes (boxcars) being the highest. There are exactly 36 possible combinations of pips, as show in the chart below.

The probability of any one of these specific rolls coming up is exactly 1 in 36, or 2.78%. The probability of any specific total coming up is the sum of the probabilities of each way it can be made. When making wagers at the Craps table, these figures can be used to forecast the likelihood of a winner (or loser). They can also be used to better understand how payouts and odds interrelate.

Probabilities and Payouts

For example, there are six combinations that create the total 7, so the chances of a 7 being rolled are 6 in 36, which is equivalent to 1 in 6 or 16.67%. The payout for a unit bet on Any Seven is 4-to-1, significantly lower than the true odds, and giving the House its advantage. Specifically, if a bet is made on Any Seven for 36 straight times, it can be expected to win six times, paying 24 units, but it will lose the other 30 times for a net loss of six units.

Similar calculations can be made for Any Craps, Yo (11), and C&E bets. As the chart indicates there are just four combinations that result in craps (2, 3, or 12). Simple math show the probabilities for Any Craps as 4 in 36 or 11.11%. For a Yo-leven (11), the result is 2 in 36 or 5.56%. The probability of a C&E bet winning is the sum of these probabilities, namely 6 in 36, or 16.67%–the same as Any Seven.

On a split C&E bet, a craps pays 3-to-1 and an 11 pays 7-to-1. Again using 36 trials, there should be four craps winners of three units each and two Yo winners of seven units, for a total of 26 units won. However, the wager will lose the other 30 times, for a net loss of four units. As the numbers indicate, C&E is a slightly better wager than Any Seven, even though the probabilities are the same.

Other Examples

Because the House has an advantage on every betting area of the table, the player’s best strategy is to use combinations of wagers to try and tilt the odds in his/her favor. For example, Place Bets on the 6 and 8 will cover 10 of the 36 combinations, which is a 27.78% chance of success. The payout is 6-to-5 each time one of these numbers comes up. But Place Bets are not single roll bets. They only lose when the number 7 appears, which has a 6 in 36 or 16.67% chance of being thrown on any roll.

This explains why Place Bets are so popular. In the short term, they can tip the odds in the player’s favor, even though the House edge will prevail over the long term.

On the other hand, Hardway Bets are also very popular, even if the probabilities don’t encourage an enthusiastic following. Each Hard number has only one winning combination—a 1 in 36 chance of success. But again, these are not one roll wagers. To calculate the true odds of winning, one must consider the number of ways of losing.

For the Hard 4 or Hard 10, the chart shows eight ways to lose—six combinations that make 7 and two Easy totals. In other words, the probability of winning is not 1 in 36 for these bets, it is 1 in 9, and the payout is 7-to-1. For the Hard 6 and Hard 8, there are ten ways to lose—the same six combinations of 7 plus four Easy totals. The chance of winning on these bets is 1 in 11, and the payout is 9-to-1. The attraction of Hardway wagering is the long odds, not the probability of success.