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# Gaming Guru

### The Pass Line

1 September 2003

I am sure that you are all familiar with the house advantages of the standard casino games: 5.28% for Roulette, 5.22% for Caribbean Stud, 2.85% for Let it Ride, 3.37% and 2.32% on the Ante/Play wager and the Pair Plus wager in Three Card Poker, and 1.41% for the Pass Line in Craps. These figures are regularly tossed around by gaming writers. But how do folks arrive at these numbers? This question in regards to Let It Ride was addressed by me in two articles on this site back in May and June of 2001. I may address some of the other games in the future. But Craps? Everyone understands that game. Or do they?

I recently received a question from a gaming writer in Mississippi asking me to explain how one calculates the 1.41% figure for the Pass Line in Craps. In the course of our correspondence, and from other letters I have received, I realized that although most players know the 1.41% figure, many do not really understand how to calculate it. In this article I want to explain how to calculate it in what I hope will be a clear and concise manner.

To begin with, the magic number is 1980. How do I come up with 1980? Since there are 36 different outcomes when a pair of dice is rolled (it helps to think of each die as being a different color), I want a number that is divisible by 36. The rest of it is hindsight. In the course of our calculations, you will see that I also need a number that is divisible by both 5 and 11. Hence, the number I use is 36 x 5 x 11 = 1980.

Very well, let us suppose that we play the Pass Line 1980 times and everything happens exactly as probability theory predicts. On each comeout roll here is what we can expect to see: 1/36 of the 1980 rolls, or 55 of them, will be a total of 2; 2/36 of them, or 110, will be a total of 3. And so on. Here is a table showing the breakdown.

 Dice Total Frequency 2 55 3 110 4 165 5 220 6 275 7 330 8 275 9 220 10 165 11 110 12 55 Total - 1980

So a natural, a 7 or 11, will be rolled on the comeout 330 + 110 or 440 times. Craps, 2,3, or 12, will be rolled 55 + 110 + 55 or 220 times. The rest of the comeout rolls are points.

Since there are 3 ways out of 36 to roll a 4 and 6 ways out of 36 to roll a 7, there are only 9 of the 36 possible dice outcomes that settle the point of 4. This means that 1/3 of the time the 4 will be rolled before the 7 and 2/3 of the time the 7 will be rolled before the 4. Thus, of the 165 rolls shown in the above table corresponding to the point of 4, 1/3 x 165 or 55 of these will win and 2/3 x 165 or 110 will seven out. Clearly this same calculation applies to the 10 as well.

For the point of 5, there are 4 ways to roll the 5 and 6 ways to roll the 7, so there are 10 of the 36 possible dice outcomes that settle this point. So 2/5 of the time the 5 will be made and 3/5 of the time the shooter will seven out. Hence, of the 220 comeouts corresponding to the point of 5, 2/5 x 220 or 88 of them will be a 5 made and 3/5 x 220 or 132 of them will be a seven out. See, I told you we needed a number divisible by 5.

For the 6, the numbers are 5/11 x 275 or 125 and 6/11 x 275 or 150. Now all of this information can be tabulated and combined with payoffs as follows:

There you have it. For 1980 Pass Line bets, say \$1 for each, you would, on average, lose \$28. Thus, the house edge is 28/1980 expressed as a percentage:

House Edge = 100 x 28/1980 = 1.41414141…%

This article is the first of four on the game of Craps. Next month I'll look at the consequence of taking Odds on a Pass Line bet. See you then.

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Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers