Craps Odds

The House Edge

Like roulette, it is fairly simple to calculate the house edge on the bets, at least most of them. Start with the proposition that each die has six numbers, each with a 162/3% probability of turning face up on a throw. The chance of any particular combination of two dice, then, is one-sixth time one-sixth, or one thirty-sixth. In percentage terms, this is 2.78%. The frequency of rolls is symmetrical among the possibilities:

The odds of a natural on a Come Out roll are therefore 8 out of 36 or 22.22% (2 in 9, or 7:2 odds). The odds of craps on a Come Out roll are 4 out of 36 or 11.11% (one in 9, or 8:1). So the odds of ending the game on the Come Out roll are 12 possibilities out of thirty-six, or 1 in 3. The other two-thirds of the time, there will be a Point. When the Point is “made,” the player wins. The odds of rolling the point, given the come out roll (i.e, not including that possibility), depend on what the point is.
If the Point came out as 4, the odds of throwing it were 3 in 36, or 1 in 12. The odds of throwing it again are the same, but the odds of throwing a 7 are twice as good: 6 in 36. By the same token, the odds of throwing a 10 are the same as for four.

Working out and summing all the possible outcomes is a tedious task, but doable. The final answer is that a person has overall slightly less than a 50-50 chance of winning the Come Out roll. As the bet is paid 1:1, the house has an edge. That edge is less than 1 percent, barely. In fact it is -0.925% under most Las Vegas craps rules.

Recall that the “Free Odds” bets are at zero edge. So if a Pass Line bet is followed up by a much higher Odds bet, the average edge of the casino on that one wager transaction is diluted. An Odds wager of 20 times the Pass Line or Come Line wager would dilute the house edge to only one tenth of one percent.

The “Don’t Pass” bet is about 0.15% worse for the house than the Pass Line bet, mainly because of the push on boxcars. By combining a “Don’t Pass” and a heavy Free Odds bet, the house edge can be almost obliterated.

“Place” bets are less generous than the Pass Line or Come bets. For example, the odds of winning a place bet on a 6 are 5 chances out of 36. The odds of losing the place bet (with a seven) are 6 chances out of 36 and the odds of a push on one roll (something other than the point or a 7) are 25 chances out of 36. So the “real” expectation of a single throw is 13.89% for a win, 16.67% for a loss, and 69.4% for a push. If there is a push, then another throw would follow with the same probabilities. At 7:5 odds, the financial expectation of one throw with a $12 bet is $17 * .1389 + $0 * .1667 + .6944 * 12. The answer is $10.69, clearly a negative bargain at around a -10% edge for the house. One might think, “Yes, but there is another roll of the dice in the case of a tie, so the bet is not over yet.” The problem is that the expectation of the next roll is the same as the first one, that is, it is negative. It is not possible to sum a series of negative-expectation bets to come up with a positive one. Even taking this nested betting proposition to a mathematical limit does not turn it into a positive one.

Field Bets are single roll bets. If a 2 is rolled, the payout is 2:1 (with a 35:1 probability), and if a 12 is rolled, the payout is 3:1 (with a 35:1 probability), and if 3, 4, 9, 10 or 11 is rolled, the payout is 1:1. As there is only one roll and one payout, the calculation needs to take into account the compound possibilities. The overall chance of winning something is 16 out of 36, or 44.4%. The weighted average payoff (taking into account the likelihood of each outcome) is 35:19, which is little worse than 2:1. Calculating the payoffs and likelihoods of each outcome and summing them up yields a house edge of -2.8%.

Proposition Bets are fairly straightforward to calculate. Craps pays 8:1. The real odds of craps are 4 chances out of 36, or 1 in 9, or 8:1. So this is an even bet. Sevens pay 5:1. The real odds of a seven are 6 chances out of 36, or 5:1. So this is also an even bet. Elevens pay 16:1, with real odds of 2 out of 36, or 1 out of 18, or 17:1. This bet has an edge for the house. The edge is -5.6%. A 3 will be paid also at 16:1. The chances of a three are the same as for an 11, so the bet is the same. Aces or boxcars pay 30:1. The real odds are 1 out of 36 for either of them, or 35:1. This bet also has an edge for the house. It is -13.89%.

The Horn bet is a collection of 4 negative proposition bets, so its edge will be roughly -10%.

Hardways is a little tricky to calculate, as it involves multiple rolls, like the place bets. A single roll will render the “hardway” one in 36 times and the “easy way” either 3 times in 36 or 5 times, depending on whether it is a 4 or 10 on the one hand, or a 6 or 8 on the other. On a single roll for a double 5, going for a 10 the hardway, the odds of getting the double are 1 in 36 to win, and 3 times for the easy way (which loses) and 6 times for a seven (which also loses) and the rest is a push. The probabilities are: to win, 2.77%, to lose, 25%, and to push, 72.22%. At 10 to 1, the player has an edge of approximately 2.78%. At 9 to 1, the edge is zero.

Most experienced players recommend starting with Pass Line bets, followed up with Full Odds bets. Stay away from the place bets, field bets and propositions for a while. Played like that, a bankroll has a better chance of lasting the evening.