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pwcrabb
pwcrabb
Joined: May 15, 2010
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August 16th, 2019 at 5:50:38 PM permalink
DeMango gets full credit for his subtle lowbrow Ad Hominem argument. I agree with Steen that the Net Present Value approach to evaluating wagering futures has seldom appeared in print.

One potentially useful nugget for Steen is the observation is that the next Big Red, with its probability each roll of ( 1 / 6 ), and with its entire future evaluated only in a constantly changing NOW, can be projected as likely to appear after 3.801784 future rolls. Likelihood is defined as probability greater than ( 0.5 ). As each roll evolves from future to past, it becomes irrelevant to the analysis and is discarded.

A corollary to this observation is the implication that half of all future Reds will require from one to 3.801784 tosses to appear and the other half will require more tosses, with no upper limit. If we use the approximation of four tosses then we are including roughly 51 % of all future Reds. This forecast in no way invalidates the fact that overall the average non-Red interval is five tosses. One may easily evaluate a sizeable sample of Reds to test whether the sample divides nearly equally into "short" and "long" types.

Can this division of future Reds into two equally likely types be useful? It can be useful if we consult clustering theory, which describes the clusters in which events of defined probability are likely to be distributed across future time. In a simple problem like this binary model, for a selected cluster size ( N ) of favorable outcomes, e.g. nine consecutive "short" Sevens requiring four or fewer rolls to appear, the proportion over infinity of all future Sevens that is represented by clusters of size ( N ) is equal to:

{ ( N ) / [ ( 2 ) ^ ( N + 1 ) ] }

In practical terms, such clustering of "short" and "long" Sevens is variously experienced as Craps tables that are "cold," "hot," or "choppy." One can state affirmatively that such clusters will in fact appear and in what proportions. However, no one can rationally forecast the crucial matter of WHEN they will appear.
Last edited by: pwcrabb on Aug 16, 2019
"I suppose I was mad. Every great genius is mad upon the subject in which he is greatest. The unsuccessful madman is disgraced and called a lunatic." Fitz-James O'Brien, The Diamond Lens (1858)
DeMango
DeMango
Joined: Feb 2, 2010
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August 16th, 2019 at 8:35:00 PM permalink
Quote: pwcrabb

DeMango gets full credit for his subtle lowbrow Ad Hominem argument.

In practical terms, such clustering of "short" and "long" Sevens is variously experienced as Craps tables that are "cold," "hot," or "choppy." One can state affirmatively that such clusters will in fact appear and in what proportions. However, no one can rationally forecast the crucial matter of WHEN they will appear.


Thank you. Just trying to keep my name out of red.
Streaks happen, you cannot predict the beginning or the end. Iím sure others will reinforce the faulty arguments.

No arguments about your faulty math?
When a rock is thrown into a pack of dogs, the one that yells the loudest is the one who got hit.
FleaStiff
FleaStiff
Joined: Oct 19, 2009
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August 16th, 2019 at 9:00:39 PM permalink
I rarely take a place bet down, I have a friend who rarely lets a place bet ride much less presses it.

I do realize that no one knows when a humungous roll is starting anymore than they know when it will be Point,SevenOut for a while.

I've never noticed much difference in our winnings.
Steen
Steen
Joined: Apr 7, 2014
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August 18th, 2019 at 8:13:10 AM permalink
Quote: pwcrabb

One potentially useful nugget for Steen is the observation is that the next Big Red, with its probability each roll of ( 1 / 6 ), and with its entire future evaluated only in a constantly changing NOW, can be projected as likely to appear after 3.801784 future rolls.


I agree there's a 50% chance of seeing a 7 within 3.801784 rolls but that seems to be a simple matter of probability and not one of Net Present Value.

As I'm sure you know, the easiest way to figure this is to take 1 minus the probability of not rolling a seven in X rolls.

1-(5/6)^X

For example, the prob of one or more 7's in 3 rolls would be:
1-(5/6)^3 = 1 - 125/216 = 42.13%


Here's a table of the first 10 rolls:

Roll pNo7% p7%
1 83.33 16.67
2 69.44 30.56
3 57.87 42.13
4 48.23 51.78
5 40.19 59.81
6 33.49 66.51
7 27.91 72.09
8 23.26 76.74
9 19.38 80.62
10 16.15 83.85


To get the 50% figure you set (5/6)^X = 0.5 and solve for X

ln = natural log

ln(5/6)^X = ln(.5)
X = ln(.5) / ln(5/6) = 3.801784

As for the unpredictable clustering, I wouldn't call that a nugget of wisdom since it's is of absolutely no use.

Steen
pwcrabb
pwcrabb
Joined: May 15, 2010
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August 18th, 2019 at 4:43:32 PM permalink
I agree with Steen that the forecast for the next appearance of Big Red is not related to Net Present Value analysis. I also agree with his method to discover the expected number of rolls to achieve 50 % probability for Big Red. That result may be used to forecast the number of future rolls that an array of Place and Buy bets may be expected to survive.

The clustering idea emerges from chaos theory. In three dimensional space, it describes the distribution of galaxies. On a two dimensional dropcloth, it describes the distribution of random paint spattering. In one dimensional time, it describes the distribution of Big Red.

Discovering the utility of knowledge of the distribution of Big Red is not intuitively obvious. In light of the pushback that has emerged thus far on this thread, I will not elaborate.
Last edited by: pwcrabb on Aug 18, 2019
"I suppose I was mad. Every great genius is mad upon the subject in which he is greatest. The unsuccessful madman is disgraced and called a lunatic." Fitz-James O'Brien, The Diamond Lens (1858)

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