Testing random

[Thomas 'PointedEars' Lahn]

Chris Angelico wrote:

> On Mon, Jun 8, 2015 at 4:23 AM, Thomas 'PointedEars' Lahn
> <PointedEars at web.de> wrote:
>> 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.
>>
>> 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  certain number is always the same; in this example, it is 
>> always 1 − 1∕9 = 8∕9.
> 
> Yes, the probability of not getting a certain number is always 8/9...
> but the probability of not getting that number for an entire sequence
> is 8/9 raised to the power of the length of the sequence

As it is for getting every number at least once, or only two, three, four, 
five, six, seven, or eight of them.  Each number has the probability of 1∕9 
of being drawn, and for a sequence of n numbers drawn, the probability is 
(1∕9)ⁿ, *no matter* which numbers are in it.

> because they are independent events that must *all* happen. […]

As I already pointed out, your reasoning is purely *intuitive*, _not_ based 
on (correct) math, and (therefore) flawed.  And intuition lets you down 
here.

The correct, mathematical reasoning is: *Because* the events are 
*independent*, the probability of the sequence does not change depending on 
the number of occurrences of an event (outcome of a draw).  That all numbers 
occur is _not_ more likely than that only one number occurs, or two, three, 
and so on.  And that does _not_ depend on how many draws you take as there 
is an infinite reservoir for each number.

I do not think I can explain it to you better, and I am not going to waste 
my precious free time to repeat myself.  Follow the reference I gave.

-- 
PointedEars

Twitter: @PointedEars2
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