Every gambler, whether occasional or serious, should have a basic understanding of the probability of winning or losing. This understanding is fundamental to maximizing your chances of winning against the casino. Walking into a casino without understanding the odds of winning or losing each bet you expect to make is the same as taking a job as a truck driver without knowing how to drive.

The key to understanding the mathematics of craps is knowing the frequency of appearance of the eleven possible total numbers—2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12— that can appear when two dice are thrown. The following chart, taken from The Facts of Craps, by Walter I. Nolan, is the best illustration I've seen of precisely how 36 different combinations of the dice can produce these 11 numbers.

Using this chart, it is easy to compute the true odds for any given situation. First, we determine the probability of each number being thrown. For example, there are 6 ways to roll a 7, according to our chart. That leaves 30 ways that a 7 will not show; therefore, the odds are 30 to 6, or 5 to 1, that you will not throw a 7 in one roll. Similarly, there is only one way to roll a.2, as compared to 35 ways to roll some other number; thus the odds are 35 to 1 that you will not throw a 2 in one roll. The next table shows the probability of each of the 11 possible numbers appearing on the next roll.

Now we can see the odds against throwing a natural 7 (5 to 1) or 11 (17 to 1); and a craps 3 (17 to 1), 2 or 12 (35 to 1). How about our chances of making a point number after it has been established? This can be determined by

comparing the number of ways the point can be made to the number of ways to roll a 7. For instance, if our point is 4, Nolan's probability chart shows it can be rolled 3 ways, as compared to the 6 ways to throw a 7. Therefore, the odds against making a 4 before a 7 are 6 to 3, or 2 to 1. By computing the true odds for every betting situation, and then comparing them with the payoff odds offered by the casinos, we can arrive at the precise casino advantage for every bet. The mathematics are not difficult, but they are tedious. The formulae are detailed in The Casino Gambler's Guide, by Allan N. Wilson and in The Theory of Gambling and Statistical Logic, by Richard A. Epstein. With the help of these two recognized leaders in the mathematics of gambling, we arrive at the chart on page 78.

Don't bother to memorize the percentages in this chart, since as a smart player you will be making only a few of these bets, studiously avoiding any wager with a casino edge approaching 2% or more. Making wagers which pay off with a disadvantage as great as almost all the bets on the chart is the quickest way to wipe out your bankroll. The only real difference between an astute craps shooter and a mark is the knowledge and use of percentages comparing the number of ways the point can be made to the number of ways to roll a 7. For instance, if our point is 4, Nolan's probability chart shows it can be rolled 3 ways, as compared to the 6 ways to throw a 7. Therefore, the odds against making a 4 before a 7 are 6 to 3, or 2 to 1. By computing the true odds for every betting situation, and then comparing them with the payoff odds offered by the casinos, we can arrive at the precise casino advantage for every bet. The mathematics are not difficult, but they are tedious. The formulae are detailed in The Casino Gambler's Guide, by Allan N. Wilson and in The Theory of Gambling and Statistical Logic, by Richard A. Epstein. With the help of these two recognized leaders in the mathematics of gambling, we arrive at the chart on page 78.

Don't bother to memorize the percentages in this chart, since as a smart player you will be making only a few of these bets, studiously avoiding any wager with a casino edge approaching 2% or more. Making wagers which pay off with a disadvantage as great as almost all the bets on the chart is the quickest way to wipe out your bankroll. The only real difference between an astute craps shooter and a mark is the knowledge and use of percentages.

When you make a regular pass-line bet at the craps table, you are playing against a casino edge of 1.414%. To figure out exactly what this means to you, estimate the total amount of bets you might make in an hour and multiply it by this figure; the result will be an average hourly cost of shooting craps. For instance, if your bets total $1,000, the casino wins $14.14. This is in the long run. In the short run, which could be one hour, one day, one week, one month, or even one year, you may be on the winning side, or you may lose more or less than 1.414%. Every gambler walking into the casino believes he is the lucky one who will beat the house percentage. Sometimes you do, but more often you don't.

There are some astute craps shooters who call the lower casino advantage percentages for free odds an "illusion." Donald Schlesinger states:

If two people each bet exactly the same amount on the pass line, but one takes the free odds while the other doesn't, they will both lose exactly the same amount of money (1.41% of the pass-line action) in the long run. The lower percentages above are always working on a larger bet than the player originally intended to make, thus the "illusion" of getting more for your money. In reality, when you stop to think of it, there is really no benefit at all where single or double odds are offered!

What I question about this reasoning is the phrase "larger bet than the player originally intended to make." I strongly believe that all bets, from the smallest to the largest, should be based on the player's bankroll and betting If you have a $1,500 bankroll and you bet three

units on the pass line at:

$1. per unit at a maximum-double-odds game, the casino advantage will be .500.

$1.09 per unit at a double-odds game, the casino advantage will be .606.

$1.42 per unit at a maximum-single-odds game, the casino advantage will be .740.

$1.52 per unit at a single-odds game, the casino advantage will be .848.

$2.53 per unit and take no odds, the casino advantage will be 1.414.

Since units of 3 are the most advantageous when taking odds, round these figures off to a $3 base bet at the double-odds game, a $5 base bet at the single-odds game, and a $7 base bet if you do not take the odds. If, however, you do not vary your bet size for the same bank according to the game you play, then the comments above about the "illusion" of an advantage are correct. The $1,500 bankroll used in this discussion is quite conservative, and, of course, you may use a smaller amount. The important thing to remember is that you must vary your bet sizes according to the type game you play for the reduced casino advantage to be effective.