# Craps Probability

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Craps is one of the easier casino table games to understand mathematically. It is played with two cube-shaped dice, the six faces of each being numbered with “pips” from one to six. When the dice are tossed and come to rest, the combined number of pips showing on the uppermost faces of both dice is the total for the throw. This must be a number between two (1-1 or “snake eyes”) and twelve (6-6 or “boxcars”).

When wagering at the Craps table, probabilities can be used to predict how likely a winner or loser might be. They also aid in understanding the relationship between Craps odds and payouts.

In total, there are just 36 possible combinations of pips—less than the number of possible outcomes on a Roulette wheel. The probability of any specific combination being rolled is exactly 1 in 36, or 2.78%. The probability of any specific total appearing is the sum of the probabilities of each way the total can be formed.

For example, six combinations result in a total of seven (1-6, 2-5, 3-4, 4-3, 5-2, 6-1), so the chances of a seven being rolled are 6 in 36. That is equivalent to 1 in 6 or 16.67%. Because the payout for a single unit wagered on “Any Seven” is 4-to-1, the House has a significant advantage, paying far less than true odds, which would be 6-to-1. If a player were to wager on Any Seven 36 times, six wins could be expected for a payout of 24 units. However, the bet would lose the remaining 30 times, resulting in a net loss of 30 – 24 = six units.

This method of calculation can be applied for other wagers, such as Any Craps, Yo (11), and C&E bets. There are just four combinations that result in craps (1-1, 1-2, 2-1 and 6-6). That makes the probability of for Craps 4 in 36 or 11.11%. For a Yo (11), there are just two ways to win (5-6, 6-5), so the probability is 2 in 36 or 5.56%. For a C&E bet, the sum of these combinations can be used, 4 + 2 = 6 in 36, or 16.67%, which just happens to be the same as for Any Seven.

When making a split bet on C&E, any craps will pay 3-to-1 while rolling eleven will pay 7-to-1. In 36 rolls, there should be an average of four craps winners worth three units each and two Yo winners worth seven units, making a total of 26 units in winnings. The wager will lose, however, the other 30 times, resulting in a net loss 30 – 26 = four units.

As the numbers show, wagering on C&E is marginally better than betting on Any Seven, even though the probabilities are the same. That because of the way the payout ratios are set to give the House its advantage. In fact, because the House has a built-in edge for every betting area on the table, the best strategy for the player is to combine wagers in ways that tilt the odds toward obtaining some kind of win on virtually every roll.

Take, for example, making Place Bets on the box numbers 6 and 8. Together, they cover 10 of the 36 combinations for a 27.78% chance of success. Each time one of these two numbers appears, the payout is 6-to-5. However, Place Bets are not wagers on a single roll. They lose only when a total of seven is rolled, which occurs at a rate of 6 in 36 or 16.67%. If a player can win twice on the Place Bets before a seven come up, a profit is assured.

For this reason, Place Bets are very popular among Craps players. The House edge will always prevail in the long term, but at least over the short term, Place Bets offer a way to flip the odds in the player’s favour.

By contrast, Hardway Bets are also quite popular in spite of poor probabilities of success. Each Hard number has just a single combination, which means a 1 in 36 likelihood of winning. Because these are not one roll wagers, the player is hoping that wins will occur in sufficient number to mitigate losses when the seven shows up. In order to determine the true odds of success, it is necessary to calculate the number of ways of losing.

For the Hard 4 or Hard 10, there are eight ways to lose—six combinations that make seven (1-6, 2-5, 3-4, 4-3, 5-2, 6-1) and two Easy totals each (1-3, 3-1 and 4-6, 6-4). In other words, the probability of winning is not 1 in 36 for these bets; it is actually 1 in 9, although the payout is only 7-to-1, well below true odds.

For the Hard 6 and Hard 8, there are ten combinations that produce a loss—the same six combinations of 7 plus four Easy totals each (1-5, 2-4, 4-2, 5-1 and 2-6, 3-5, 5-3, 6-2). The probability of winning on these bets is thus 1 in 11, and the payout is 9-to-1. Obviously, for those who understand the math, Hardway wagering is attractive only because of the long odds offered, not the likelihood of winning.

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