Sanderson M. Smith

CRAPS ... A CASINO GAME OF PURE CHANCE

Craps is an exciting game with lots of action. If you want to try to double your money quickly on a game of pure chance, one of your best opportunities is to bet all of your money on one game of craps. Your probability of winning is just slightly under 50%, as will be shown below.

Let me quickly state that I am talking about straight craps, and not the multiple number of side bets you can make while someone is rolling the dice. The house has only a slight advantage on straight craps, but has a large advantage on the side bets.

Craps involves the rolling of two dice. Here are the rules:

If you roll a total of 7 or 11 on the first roll, you win.

If you roll a total of 2, 3, or 12 on the first roll, you lose.

If you roll a total of 4, 5, 6, 8, 9, or 10 on your first roll, this number becomes your point. You continue to roll the dice. If you get your point total before a total of 7 appears, you win. If you roll a total of 7 before your point total appears, you lose.

Note: This is a 1:1 bet. That is, when you win, you win a dollar for each dollar that you bet.

The possible totals obtained from rolling two dice are shown at the right. Note that there are 36 cells containing the totals, and each cell has a probability of 1/36 of being the result of a craps roll.

 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12

First off, note that you can win on the first roll. Your probability of getting 7 or 11 is 8/36, or about 22.22%. You can also win if you roll a total of 4, 5, 6, 8, 9, 10. The winning possibilities will be summarized below, but let's first look at an example of winning with a point total. For instance, what is the probability of winning if you initially roll a total of 5? This involves

• rolling of 5 on the initial roll, and then
• rolling another 5 before you roll a 7.

OK, the probability of rolling a 5 is 4/36. Once you have rolled the 5, the only cells (check table above) that matter are the ones containing 5 and 7. Every other cell can be ignored. There are ten cells containing 5 or 7. Four of these cells are "favorable" to you. The probability that you will roll a 5 before a 7 is 4/10. Hence, the probability of rolling a 5, and then rolling a 5 before a 7 is (4/36)(4/10), or about 4.44%.

Here is a summary of your winning probabilities in craps:

 YOUR INITIAL TOTAL PROBABILITY OF WINNING PROBABILITY WRITTEN AS DECIMAL 4 (3/36)(3/9) 0.027778 5 (4/36)(4/10) 0.044444 6 (5/36)(5/11) 0.063131 7 6/36 0.166667 8 (5/36)(5/11) 0.063131 9 (4/36)(4/10) 0.044444 10 (3/36)(3/9) 0.027778 11 2/36 0.055556 TOTAL 0.492929

In craps, if you are not the roller, you can bet with the roller (who is called the shooter) by putting your money on the Pass Line. You are making what is known as a Pass Bet. As the table indicates, the probability of winning a Pass Bet is 49.29%.

The casino (sometimes called the house) has a 50.71% probability of winning. Hence, the casino has a 1.4% advantage on any Pass Bet.

Interestingly, it is possible for you to bet with the casino, and against the shooter. To do this, you put your money on the Don't Pass Line. However, if you place a Don't Pass Bet, and the shooter initially rolls a 12, the casino wins, but you don't. You don't lose your bet, but you don't win anything. The probabilities associated with a Don't Pass bet are as follows:

 Event Probability TIE on Don't Pass bet. 1/36 = 0.027778 WIN on Don't Pass bet 0.507071 - (1/36) = 0.479293 LOSE on Don't Pass bet 1 - (1/36) - 0.479293 = 0.492929

Now, on the tie, nobody wins. So, let's ignore the tie situation. If we reduce this to a win/lose situation, the probability that a Don't Pass Bet wins is .479293/(.479293 + .492929) = .492987 and the probability that a Don't Pass Bet loses is 1 - .492987 = .507013. Simple subtraction shows that by barring the 12 total to the Don't Pass bettor, the casino maintains a 1.4% advantage over the player.

The casino has only a slight edge in craps. But, in the long run, the game is a money-maker for the casino since the casino plays on indefinitely, and needs only the slightest advantage to "come out ahead." And, the casino managers are delighted when craps players bet on the many other various options associated with the game. From the player's standpoint, these options are not good bets.

You won't make a living playing craps at a casino. About all that can be said is that you won't lose your money quite as fast as you will in other casino games of pure chance.