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Best of Donald Catlin

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The Number of Craps Rolls

1 June 2013

In January and February of the year 2000 I wrote a two part article dedicated to calculating the average length of a player’s Craps hand in terms of the number of rolls. The answer turns out to be a figure of about 8.526. Recently a reader by the name of Martin K wrote a letter to me challenging this figure, part of which I’ll quote below.

"I currently play at a casino (Maryland Live) that has a craps game (Interblok) that uses an analog set of dice and a digital interface (no dealers, sticks, etc.) It keeps tracl of the number of rolls and displays it over the game. I have played it about a dozen times (or more) and the average number of rolls per line is consistently about 4…. In addition and before this I had developed a number of computer simulations that tracked the number of rolls on a point-only basis and these have consistently shown the average length to be about 4 counting the point being set as 1 and the resolution of the point as the end. If I include the trash rolls it shows about 3.5 which is consistent with MD Live….

"Consequently, while the math shows 8.5 as the average the observations do not bear that out. What gives?

"Martin
"Systems Engineer – Global Positioning System"


Well Martin, I’m not sure what gives but I do know that the 8.526 figure is correct. I’m not sure what you mean by a “point-only basis” nor do I understand what setting the point as 1 means. You need not explain though because I do know something is wrong. Rather, I thought that the easiest thing to do to convince you that the mathematical analysis is providing the correct information is to write a short simulation in True Basic that confirms it. I have provided that below:

RANDOMIZE
FOR K = 1 to numhands
IF INT(K/100000) = K/100000 THEN PRINT k
LET nr = 0 !Initialize player’s hand !nr is the number of rolls in the hand
DO
CALL ROLL(tot) !Comeout roll
LET nr = nr + 1
IF tot = 7 OR tot = 2 OR tot = 3 OR tot = 11 OR tot = 12 THEN
LET tot = 0 ! Pevent interpreting 7 as a missout
ELSE
LET point = tot
DO
CALL ROLL(tot)
LET nr = nr + 1
IF tot = point OR tot = 7 the EXIT DO
LOOP
END IF
IF tot = 7 then EXIT DO !Ends hand
LOOP
LET gtot = gtot + nr !Running grand total
NEXT k
LET ar = gtot/numhands
PRINT “The average number of rolls per hand is”;ar
END

SUB ROLL(tot) ! Rolls dice – tot holds dice total
LET d1 = INT(6*RND + 1)
LET d2 = INT(6*RND + 1)
LET tot = d1 + d2
END SUB

If you run this program using a hundred million trials you should find that the variable ar is a number between 8.52 and 8.53. I hope this clears things up for you. See you all in a couple of months.

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