Basic Probability

This is a discussion on Basic Probability within the The Foyer forums, part of the Off the Grid category; The underlying purpose of this thread is to examine the psychology of defence. I am primarily interested in the mechanisms ...

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Old Mar 28, 2009, 8:10am   #36
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TheBramble started this thread The underlying purpose of this thread is to examine the psychology of defence. I am primarily interested in the mechanisms that are employed to defend established dogma.

Not the longer term members who know me well enough to have seen me coming, but there’s a new crop of ‘stars’ out there who have yet to be put through the mill and whose insights and fresh angles could be potentially significant. Even more so, on a personal level, their approaches and styles of response.

Although I appreciated the majority, the vast majority would not for a second consider the possibility that a review of any nature on this topic would be appropriate or necessary, there are, as always, two camps.

Of those who felt it was beyond question, few, if any, would offer a logical basis of proof for supporting existing beliefs other than to simply regurgitate the self same laws which are apparently being scrutinised. Although it was to be expected that the majority would be those resistant to any form of challenge, I am fairly sure the few who did stop to consider the question afresh, however briefly, would do so in anonymity and would be unlikely to stick their head above the parapet to disclose their momentary heresy. There is a visceral biological imperative to staying small and keeping quiet in the early stages of any major shift.

Of those who considered my thinly disguised intentions (post #16) impertinence, there would be the usual crop of blowhards who would feel it was an attack on them personally and respond in kind. I could have nailed their names to the mast ahead of time. LOL. You weren’t targeted, you arrived here of your own free will and you participated in your own way. No blame.

I’m not sure whether it is the sense of being personally attacked (questioning current knowledge and beliefs which, for all of us, are potentially traumatic) or of having an external belief (that which we believe ‘everyone’ believes) brought into question. Which do we defend most strongly?

It’s a measure of intellectual development the extent to which we are not only willing, but eager to have the opportunity to robustly review that which we currently hold to be Fact or Truth – however that opportunity presents itself to us. The more solid, long-standing, obvious or sacrosanct the belief the more powerfully should we seize these opportunities to re-examine the fundamentals upon which they are thought to be based, the proofs and processes.

For those still up for it, stick the 16 Permutations in a word processing document and string count the occurrences of ‘HH’ and forget for a moment you know anything about probability.

Of 4 coin flips, what is the probability you will encounter a string of two consecutive Heads?
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Old Mar 28, 2009, 10:17am   #37
 
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I think you have established that there are 12 consec Heads (and Tails).
I think that sometimes we misread the intention of a question, and start answering it before we reach the end of the question.

The effort to manually go through a small exmaple, and then deduce a general property is an exercise conducted by those who are bored, or love puzzles.
(you may find puzzle-addicts are more open-minded to new ideas, as its the "Aha!" moment, or the discovery that is the prize, not re-inforcement of pre-concieved ideas.)

Of the 16 perms;
4 Hs. (HHHH) positions (1,2), (2,3) (3,4) = 3
3 Hs. positions (123, 124, 134, 234) (12, 23, 12, 23, 34, 34) = 6
2 Hs. positions (12, 13, 14, 23, 24, 34) (12,23,34) = 3
1 Hs. positions (1,2,3,4) = no consecs.

Where do we go from here?
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Old Mar 28, 2009, 10:51am   #38
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TheBramble started this thread
Quote:
Originally Posted by trendie View Post
you may find puzzle-addicts are more open-minded to new ideas, as its the "Aha!" moment, or the discovery that is the prize, not re-inforcement of pre-concieved ideas.
Ah. Nail on the Head. Not inappropriately either for a student of Maslow…

Not sure if it’s that evenly divided between just two camps, but you may be right.

Quote:
Originally Posted by trendie View Post
I think you have established that there are 12 consec Heads (and Tails)
Where do we go from here?
OK, bearing that figure in mind and the formal statistical probability for the same, if you were offered the opportunity to take part in a high-stakes wager where you win on the first occurrence of two consecutive Heads in a maximum 4 coin tosses game, what would your Aw:Al ratio have to be for it to become marginally positive enough for you to consider participation?
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Old Mar 28, 2009, 6:12pm   #39
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TheBramble started this thread OK. Enough is enough.

The Probability of two consecutive Heads in 4 coin tosses is derived from the formula (Gardner, Berlkamp below)

1-(F(n+2)/(2^n))
where F(n+2) is the (n+2)th Fibonacci number

In our example the value is 0.1875 (3/16).

For the last question asked, the probability of two consecutive Heads OR 2 consecutive Tails the result is a Probabilistic OR which yields 0.375 (6/16).

These are ‘real life’ probabilities. Ones that actually occur. The ones YOU WILL experience regardless of what the textbooks say. These apply to us, to our trades and our trading performance statistics and all the other ratios associated with trading performance over time.

That’s why I kept referring you to the 16 Permutations in my earlier post. They are the least likely set of strings you’ll ever find in 16 sets of 4 coin toss results. They are idealised outputs from a Gaussian Universe that will never be encountered in any reality you inhabit, accidentally or otherwise. Yet they look ‘correct’. That’s what you’d expect from elementary probability theory – and you come to expect it and feel comfortable with it. There’s the real danger.

If you carry on trotting out, or even worse believing, and even worse still, financially depending upon all the other textbook stuff on Probability which applies to a Gaussian Universe you will potentially come a cropper. Just like LTCM did. The real Universe of Coin tosses (and trading) has fat tails AND skew.

For any who have the vaguest hint of energy or interest left in this topic, have a shot at guessing the minimum coin tosses you’d need to make to, statistically, in our very real, but non-Gaussian Universe, generate all of the 16 possible strings of results from a nominal 4 coin toss sequence?

If you don’t have any idea, set a lower bound and an upper bound for the numbers you think are 90% likely to encapsulate the actual answer to this one. I’ll post the answer later.









Gardner, M. "Nontransitive Paradoxes." Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 64-66, 1988.

Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 1: Adding Games, 2nd ed. Wellesley, MA: A K Peters, p. 777, 2001.

Last edited by TheBramble; Mar 28, 2009 at 6:47pm. Reason: typo
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Old Mar 28, 2009, 6:27pm   #40
 
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TTTT
TTTH
TTHT
TTHH
THTT
THTH
THHT
THHH
HTTT
HTTH
HTHT
HTHH
HHTT
HHTH
HHHT
HHHH

I get 8 out of 16 will give 2 consec heads!

EDIT: actually, with the exception of HTHT and THTH, ALL other options will give at least one consec, be it Heads or Tails.
So, 2/16 will not give a consec, thus 14/16 options will give a consec Head or Tail.
__________________
# If the only tool you have is a hammer, you tend to see every problem as a nail - Abraham Maslow
# There are 10 kinds of people in the world; those that understand binary, and those that dont. -Anon
# Ed Seykotas Whipsaw Song http://www.youtube.com/watch?v=LiE1V...Wlxk8&index=10
# Defeat is temporary. Giving up makes it permanent. Anon

Last edited by trendie; Mar 28, 2009 at 6:36pm.
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Old Mar 29, 2009, 11:28am   #41
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TheBramble started this thread So, based on what you think it currently is trendie, if you were offered the opportunity to take part in a high-stakes wager where you win on the first occurrence of two consecutive Heads in a maximum 4 coin tosses game, what would your Aw:Al ratio have to be for it to become marginally positive enough for you to consider participation?
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Old Mar 29, 2009, 11:57am   #42
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Guys, probabilities make sense in your examples only as relative frequencies. There is a lot of debate about the relevance of of limiting probabilities to single or few step decision making processes.
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