SteveSak
Joined: Aug 28, 2015
• Posts: 1
August 28th, 2015 at 10:44:42 AM permalink
This is with a doubt my absolute favorite gambling side. What a tremendous wealth of information!!

I would like to know how the number of decisions per hour in craps is calculated. I have typically seen a number that was 30% of the the rolls per hour.

What is the math behind arriving at this 30% number?

Steve in Newbury Park
MathExtremist
Joined: Aug 31, 2010
• Posts: 6526
August 28th, 2015 at 11:08:31 AM permalink
There are about 3.376 rolls per passline decision, on average; 1/3.376 is about 30%. The math behind the 3.376 number is a good exercise though -- do you know how to calculate it?
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563
mustangsally
Joined: Mar 29, 2011
• Posts: 2463
August 28th, 2015 at 11:27:58 AM permalink
nice question for fun math (math IS fun)
Why you want to know the math?

ok
Donald Catlin (such a nice man, imo, and funny too, like Ed Sullivan funny)
5 January 2000
wow!

http://catlin.casinocitytimes.com/article/how-long-is-a-craps-roll-1240

First let us calculate the expected number of rolls for a Pass Line decision.
To do this we will calculate the expected number of rolls for each of the possible things that will lead to a Pass Line decision and weight each one by the probability that it occurs."

EventProbabilityExpected RollsProduct
Craps/Natural 12/361 12/36
Point 4 3/361 + 36/93/36 + 3/9
Point 5 4/361 + 36/104/36 + 4/10
Point 6 5/361 + 36/115/36 + 5/11
Point 8 5/361 + 36/115/36 + 5/11
Point 9 4/361 + 36/104/36 + 4/10
Point 10 3/361 + 36/93/36 + 3/9
Totals1.1 + 392/165

so that 30% is close to 165/557
i due agree

now the distribution
the chance that on roll X (or by roll X) the craps game ends (round - pass line is resolved)
1/3 chance on the 1st roll or the come out roll
1 in 15 that the round requires more than 8 rolls
I have seen 20 rolls to a decision. point was 4 and it lost.
rollends on Nth rollends by Nth rollstill no decision1 in
00011
10.3333333330.3333333330.6666666671.50
20.1882716050.5216049380.4783950622.09
30.1347736630.6563786010.3436213992.91
40.0965673110.7529459110.2470540894.05
50.06925710.8222030110.1777969895.62
60.0497177150.8719207270.1280792737.81
70.0357251280.9076458550.09235414510.83
80.0256953610.9333412160.06665878415.00
90.0184993250.951840540.0481594620.76
100.0133314870.9651720270.03482797328.71
110.0096166450.9747886730.02521132739.66
120.0069437020.9817323740.01826762654.74
130.0050185750.986750950.0132490575.48
140.0036307030.9903816530.009618347103.97
150.0026291790.9930108320.006989168143.08
160.0019057530.9949165850.005083415196.72
170.0013826970.9962992820.003700718270.22
180.0010041490.9973034320.002696568370.84
190.0007299220.9980333540.001966646508.48
200.0005310760.998564430.00143557696.59
210.0003867540.9989511840.001048816953.46
220.0002819060.999233090.000766911,303.93
230.0002056650.9994387550.0005612451,781.75
240.0001501750.999588930.000411072,432.67