View source for Kay (unit)

: {{Caveat|The ''Kay'' and the ''[[Kaymac]]'' are not recognised units}} but instead something I ([[User:Alec|Alec]]) have been using for years (the concept, over 7, and the name just over 1 year, as of middle of June 2018) as these units (or the values before they were named) are exceptionally useful and deserved a shorthand way to be spoken of.
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==Definition==
A ''kay'' fundamentally a unit of probabilistic rarity, the higher the number of ''kays'' an event is given by, the higher the rarity (the less common) that event is. An increase of 1 (one) ''kay'' is equivalent to the event becoming 10 (ten) times less likely, with certainty defined to be 0 ''kays''.

If an event will never occur (see: [[event has a probability of zero]], or {{XXX|Dare I say: }} is impossible {{XXX|Check terminology}}) then we cannot assign it a numerical ''kay'' value, but we can reasonably say:
* An event that will never occur has rarity: {{M|\infty}} ''kays''
** We can replace the word "rarity" with "probability" - but in this case the unit "''kays''" '''''must''''' be given
** We can also replace "rarity" with "commonality" - which is typically used for ''[[kaymac (unit)|kaymacs]]'' instead. See [[kay and kaymac values]] for how these two are related.


===Formula===
Symbolically a given [[probability value|probability]], {{M|p\in[0,1]\subseteq\mathbb{R} }} can associated to the rarity: {{M|k}} ''kays'' as follows:
* {{M|p:\eq 10^{-k} }} and
* {{MM|k:\eq -\log_{10}(p) }}, or {{MM|k:\eq \frac{-\ln(p)}{\ln(10)} }} (see [[log10 (function)]] and [[ln (function)]] for more information)
** Be aware of the warning in the note below when using {{M|\log}}, {{M|\lg}} ect in software<ref group="Note">Be aware that seldom some systems use {{M|\log}} to refer to [[natural log (function|natural log]], more commonly written {{M|\ln}}; such systems will use {{M|\text{lg} }} for {{M|\log_{10} }} instead. Because of the problems this caused most "recent" systems will use {{M|\log_{10} }} and {{M|\ln}}, consider {{M|\log}} by itself ambiguous, {{M|\text{lg} }} was unfortunately used as a shorthand for what we'd now write {{M|\log_{2} }} also called the "binary logarithm" but was used historically (especially before computers settled on using binary, rather than just representing things ''in'' binary (for example [[binary-coded decimal]], where 4 bits are used to represent the numbers {{M|0}} to {{M|9}} (and the rest wasted) saw significant use in early computers, true binary usage came later - this was especially common up until the early to mid 60s at earliest) for {{M|\log_{10} }}</ref>
==Properties==
Suppose an event {{M|\mathrm{X} }} occurs with rarity {{M|u}} ''kays'', and [[independent event (probability)|independently]] an event {{M|\mathrm{Y} }} occurs with rarity {{M|v}} ''kays'' then:
* The event of both {{M|\mathrm{X} }} and {{M|\mathrm{Y} }} occurring has rarity {{M|u+v}}

If {{M|p}} is the probability corresponding to {{M|u}} ''kays'' and {{M|q}} the probability corresponding to {{M|v}} ''kays'' then:
* {{M|u+v}} ''kays'' corresponds to the probability {{M|pq}}

This is a byproduct of ''kays'' being defined by [[logarithms]] and the defining property of logs.

 
==Table==
{| class="wikitable" border="1"
|-
! ''kays''
! Probability of occurrence
! Verbal frequency
|-
! 0
| 1 (''certainty'')
| Always
|-
! 1
| 0.1
| 1 out of 10 occurrences
|-
! 2
| 0.01
| 1 out of 100 occurrences
|-
! 3
| 0.001
| '''1 out of 1,000 occurrences'''
|-
! 4
| 0.0001
| 1 out of 10,000 occurrences; 100 in 1,000,000 occurrences
|-
! 5
| 0.00001 
| 1 out of 100,000 occurrences; 10 in 1,000,000 occurrences
|-
! 6
| 0.000001 
| '''1 out of 1,000,000 occurrences'''
|-
! 7
| 0.0000001
| 1 out of 10,000,000 occurrences; 10 in 1,000,000 occurrences
|}
==Notes==
<references group="Note"/>