Testing random

[random832 at fastmail.us]

On Sun, Jun 7, 2015, at 16:56, Thomas 'PointedEars' Lahn wrote:
> random832 at fastmail.us wrote:
> 
> > On Sun, Jun 7, 2015, at 16:09, Thomas 'PointedEars' Lahn wrote:
> >> No.  AISB, those sequences all have the same probability:
> > 
> > Yes and the probability of getting _any_ of the sequences, is the sum of
> > the probabilities for each one of the sequences individually.
> 
> The probability of getting any of the sequences is precisely 1

Any of the sequences *satisfying "contains at least one of each result"*
(or, in general, any assertion about the degree of coverage, for
example, covering at least two values being more likely than covering
only one value)

Flipping one coin has two possible sequences: H and T. The probability
of getting at least one H and at least one T is 0.

Flipping two coins has four possible sequences: HH, HT, TH, TT. The
probability of getting at least one H and at least one T is 50%, since
two of these sequences satisfies the condition.

Flipping three coins has eight possible sequences. The probability of
getting at least one H and at least one T is 75%, since six of the
sequences: THH, HTH, HHT, HTT, THT, TTH, all satisfy the condition.

Flipping four coins has sixteen possibile sequences. The probability of
getting at least one H and at least one T is 87.5%

Rolling 2d3 has nine possible sequences.
The probability of getting at least one of each value is 0.
The probability of getting at least one each of two values is 66.7%.
The probability of getting only one value is 33.3%.

Rolling 3d3 has 27 possible sequences.
Six of them have one of each value. The probability is 22.2%.
24 have at least two different values. That probability is 88.9%.
Only 3 have all of the same value. 11.1%.

Rolling 4d3 has 81 possible sequences.
36 of them have at least one of each value. The probability is 44.4%.
78 of them have at least two different values. The probability is 96.3%.
Still only 3 have all of the same value. 3.7%.

Rolling 5d3 has 243 possible sequences.
150 of them have at least one of each value. The probability is 61.7%
240 of them have at least two different values. 98.8%.
Still only 3 have all of the same value. 1.2%.


In general, as the number of trials increases, the probability of having
e.g. at least one of each value never _reaches_ 1, but it gets
arbitrarily close.