Testing random

[Thomas 'PointedEars' Lahn]

Chris Angelico wrote:

> On Mon, Jun 8, 2015 at 2:36 AM, Thomas 'PointedEars' Lahn
> <PointedEars at web.de> wrote:
>>> The greater the multiplier, the lower the chance that any element will
>>> have no hits.
>> Wrong.
>>
>>> [ex falso quodlibet]
> 
> Huh. Do you want to explain how, mathematically, I am wrong, or do you
> want to join the RUE in my ignore list?

I already did; you have overlooked it.  In a nutshell, the probability of 

  1 1 1 1 1 1 1 1 1 1 1

is *the same* as that of

  1 2 3 4 5 6 7 8 9 1 2

and the same as that of

  8 3 6 3 1 2 6 8 2 1 6.

If the set to choose from is integer numbers from 1 to 9, then *each* of 
those sequences has *the same* probability (1∕9)¹¹ ≈ 3.1866355 × 10⁻¹¹.
Repeating that random experiment one more time, the probability of

  1 1 1 1 1 1 1 1 1 1 1 2

is the same as of

  1 2 3 4 5 6 7 8 9 1 2 2

and the same as that of

  8 3 6 3 1 2 6 8 2 1 6 2

(that the next outcome is a 2): (1∕9)¹² ≈ 3.5407062 × 10⁻¹².  And the 
probability of

  1 1 1 1 1 1 1 1 1 1 1 1

(only 1s) is the *same* as of

  1 1 1 1 1 1 1 1 1 1 1 2

(only 1s and one 2) and the same as of

  1 1 1 1 1 1 1 1 1 1 1 3

(only 1s and a 3) and of

  8 3 6 3 1 2 6 8 2 1 6 4

(that the next outcome is a 4, and there are no 5s at all).

AISB, those are *independent* events; the number of occurrences of an 
outcome before *does not matter* for the probability of the next one.  And 
so the probability of getting a certain number does _not_ change depending 
on the number of times you repeat the experiment.  It is always the same;
in this example, it is always 1∕9.  And the probability of _not_ getting a 
certain number is always the same; in this example, it is always 1 − 1∕9 = 
8∕9.


So, I am sorry to tell you this, but you do _not_ understand probability.  
And you *cannot* understand it intuitively, like you tried to.  Probability 
is counter-intuitive.

That should not surprise or worry you.  Because as I and others have pointed 
out in this thread, the majority of the human population, including the OP, 
does not understand probability – until it has been explained to them.  I am 
not even sure I understand probability in every instance, or rather, I am 
sure that *intuitively* I do not.

It is why humans play the lottery using their “lucky numbers” whereas other 
numbers have exactly the same probability of being drawn; why they think a  
baby boy “is due” when they have two girls already; why they are surprised 
if two people in the room have the same birthday; why *in the short run* you 
can make a fortune by running a casino even if the tables are not rigged; 
why the “gate three” trick works (for a while); why they pay money for 
horoscopes and truthsayers, not recognizing that their wording is so 
ambiguous that it is highly probable to apply to everyone, and why many of 
them believe in a deity or fate.

A big part of the reason is psychology, how the human mind works: The human 
mind has evolved to recognize patterns.  It is an evolutionary advantage to 
be able to recognize patterns, to tell a mammoth from a sabre-stooth lion at 
a distance, a poisonous from an edible plant, and an X from a U.  So humans 
try to see patterns everywhere, even in pure randomness: clouds, texts, 
numbers, you name it.  And when they are told that something is random, and 
they find patterns regardless, they do not believe that it is random.

An extension of that misconception is emphasized by an anecdote (which may 
be apocryphal) told about Richard Feynman (I heard it from Lawrence Krauss 
in “A Universe from Nothing”; he can tell that in a much more funny way than 
I am able to reproduce it here [1]):

  Richard Feynman used to go up to people all the time and he’d say:

    “You won’t believe what happend to me today!
     You won’t believe what happend to me today!”

  And people would say:

    “What?”

  And he would reply:

    “Absolutely nothing!”

Because humans are evolutionary driven to look for patterns, and they 
believe that everything that happens to them is important and has 
significance in “the greater scheme of things” (whose mere existence
is a misconception, too).

HTH

________
[1] Lawrence KRAUSS (2009). “A Universe from Nothing”. The Mariott hotel,
    Burbank. 1:03:12.  <https://youtu.be/-EilZ4VY5Vs?t=3792>
-- 
PointedEars

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