How do casinos decide what edge to put on the different bets?

Casinos have flexibility in setting edge or advantage over players. Edge, a function not only of probability but also of payoff, measures how much the house is fiscally favored. Depending on the details of the games, it can be altered by varying either or both of these parameters.

Dice games offer simple examples. Picture a situation in which a solid citizen bets on the fall of a single standard die. A die is a cube whose six faces are identical except that they’re tagged one through six. Each of the possible outcomes – 1, 2, 3, 4, 5, or 6 – is equally likely. The chance of any result, expressed as a probability, is therefore one out of six. This can be designated as the fraction 1/6, the decimal (rounded off here to three “significant figures”) 0.167, or the percentage (also to three significant figures) 16.7 percent. Chance can alternately be stated in the form of odds, comparing ways opposing events may occur. This game has one way each number will appear and five ways it won’t; the odds against throwing any specific choice are therefore 5-to-1.

A bet on the outcome would be “fair” – no edge – if the payoff equaled the odds against winning, 5-to-1 in this case. The house would have an edge with payoffs under 5-to-1; players would have the advantage with payoffs over 5-to-1. To evaluate edge precisely, multiply the probability of winning by the payoff per dollar bet and subtract the product of the probability of losing times one dollar. This assumes that bets are returned to winners along with the payoffs. Here’s how the arithmetic works for the 5-to-1 bet when payoffs are 4-to-1 (the house has the edge), 5-to-1 (a fair game), and 6-to-1 (players have the advantage) betting on any of the six single numbers.

 
 4-to-1	(1/6)x4 - (5/6)x1 = 4/6 - 5/6 = -1/6 = -0.167 = -16.7 percent
 5-to-1	(1/6)x5 - (5/6)x1 = 5/6 - 5/6 = 0/6 = 0.000 = 0.000 percent
 6-to-1	(1/6)x6 - (5/6)x1 = 6/6 - 5/6 = +1/6 = +0.167 = +16.7 percent 

Two dice provide more outcomes and make the distribution of totals nonuniform. For each face of the first die that can finish on top, the second die can land six different ways. This leads to 6x6 = 36 possible combinations with totals from two (1-1) through 12 (6-6), as follows:

 
1-1, 1-2, 1-3, 1-4, 1-5, 1-6
2-1, 2-2, 2-3, 2-4, 2-5, 2-6
3-1, 3-2, 3-3, 3-4, 3-5, 3-6
4-1, 4-2, 4-3, 4-4, 4-5, 4-6
5-1, 5-2, 5-3, 5-4, 5-5, 5-6										
6-1, 6-2, 6-3, 6-4, 6-5, 6-6

Count the sets that total two, three, four, and so on up through 12 to verify the numbers of ways shown in the accompanying table for each, and use the tally to find probabilities. For instance, two dice total five in four of the 36 ways. The probability of rolling a five is accordingly 4/36 which reduces to 1/9 or 0.111 or 11.1 percent; quoted as odds against a five, 32 no-fives versus four fives, would be 32-to-4 which reduces to 8-to-1. A bet simply on a total of five will be fair if it pays 8-to-1; edge is with the house or the players for lower or higher payoffs, respectively.

         total   2  3  4  5  6  7  8  9  10  11  12
number of ways   1  2  3  4  5  6  5  4   3   2   1

Enquiring minds might want to know why a Place bet on five at craps pays 7-to-5, which reduces to 1.4-to-1, and not 8-to-1. Place bets on five win four ways but only lose when a seven rolls – six ways – so the odds to be overcome aren’t 32-to-4 but 6-to-4 – which reduces to 1.5-to-1. The house has an edge because it pays only 1.4-to-1. To put this in terms of probabilities and figure edge, note that four ways to win and six to lose mean 10 ways to reach a decision. The chance of winning is four out of 10 or 4/10 while that of losing is six out of 10 or 6/10. So, payoffs of 1.4-to-1 give an edge of (4/10)x1.4 - (6/10)x1 = 5.6/10 - 6/10 = -0.4/10 = -0.04 = -4.0 percent.

Sic Bo is a three-dice game with 6x6x6 = 216 combinations and totals ranging from three (1-1-1) to 18 (6-6-6). It’s easy enough to picture how many ways make a three (only one: 1-1-1), a four (three: 1-1-2, 1-2-1, 2-1-1), and a five (six: 1-1-3, 1-3-1, 3-1-1, 1-2-2, 2-1-2, 2-2-1). After that, enumerating the combinations starts getting a tad hairy, but the math mavens earn those big bucks by knowing how to use arcane calculations and computer programs to obtain the answers.

Once these figures are determined, edge can be readily ascertained for various payoffs. As an illustration, consider a bet that a five rather than anything else will occur in Sic Bo. The chance of success is six out of 216 = 6/216 = 1/36; odds beat to win will be 210-to-6 which reduces to 35-to-1. A 35-to-1 payoff would be fair. At 34-to-1, the edge would be (1/36)x34 - (35/36)x1 = -1/36 = -0.0278 = -2.78 percent. In some establishments, the payoff is 30-to-1, for an edge of (1/36)x30 - (35/36)x1 = -5/36 = -0.139 = -13.9 percent. Payoff could be and sometimes is much lower. But, then, we’d really be talking about a “sucker game.” And casino aficionados who know beans about bets will surely attest that the friendly neighborhood punting parlors they patronize wouldn’t run anything like that. At least, not in the present period of casino proliferation. For, as that cherished Chaucer of chance, Sumner A Ingmark, choicely chanted: