All Small - All Tall - All or Nothing

I recently received an e-mail from a good friend of mine who lives in Las Vegas. In it he described three new wagers that have been introduced on the Craps games at Wynn and Encore hotel casinos. The names of the three wagers are in the title of this article.

Each of the three bets can only be made after a seven has been rolled. The bets are placed in three spaces located in front of the Box Person. The bets can be from $1 to $5 and bets above $5 must be in multiples of $5. Let me describe the All Small bet for you.

The All Small bet is a wager that all of the small numbers 2, 3, 4, 5, and 6 will be rolled before the 7 occurs. If all of the small numbers occur before the 7 the player is paid 35:1 on his wager. It is the Box Person's responsibility to keep track of this wager as follows: There are five circles behind the betting area labeled 2 through 6. Every time one of those numbers is rolled the Box Person places a lammer in the circle corresponding to that number. For this wager the tall numbers are ignored. Once all the circles are covered the player wins. If the seven occurs before all of the small numbers are rolled, the player loses his wager, the circles are cleared and a new game begins.

To calculate a house edge for this wager, one must obtain the probability that all of the small numbers are rolled before the seven. One can calculate this using the same techniques that one uses to obtain probabilities for the Fire Bet, however it is not an easy calculation. I chose to do this using a simulation. I wrote a short program that randomly rolls the dice and keeps track of what small numbers have occurred; it ignores the tall numbers. As long as the seven does not show, the program keeps rolling the dice. Once the seven occurs the program stores the number of small numbers that have occurred in an array called state, wipes clean the variables that store the numbers that have occurred, and begins a new game. The program plays 500 million games. State(3), for example, records how many times three small numbers have occurred before the seven. The probability that three small numbers will be rolled before the seven is just state(3)/500,000,000.

All we really need is the probability that all five small numbers occur before the seven. Using my program I estimate this probability to be 0.026350056. With this number in hand we can fill in the following table:

As you can see the house edge for this wager is approximately 5.14%.

The All Tall wager is the same as the All Small wager except that the numbers 8, 9, 10, 11, and 12 must be rolled before the 7. The All Tall wager is symmetric to the All Small wager in terms of probabilities, so it has the same house edge.

The All or Nothing wager is a bet that all of the non-seven numbers will be rolled before the seven. If this occurs the bettor is paid 176:1. The All or Nothing wager can be analyzed in a matter analogous to the All Small analysis above except that the program must keep track of all of the numbers, small and tall. I carried out such a simulation, using a billion trials, and determined that the probability of rolling all of the numbers before the seven is approximately 0.00525391. Using this figure the house edge can be calculated as in the above table. The figure is approximately 7%.

So here is one more thing for the Box Person to keep track of. See you next month.

Don Catlin can be reached at 711cat@comcast.net