Kay (unit)

Caveat:The Kay and the Kaymac are not recognised units but instead something I (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.

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

TODO: Dare I say:

is impossible

TODO: Check terminology

) then we cannot assign it a numerical kay value, but we can reasonably say:

An event that will never occur has rarity: ∞ 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 kaymacs instead. See kay and kaymac values for how these two are related.

Formula

Symbolically a given probability, $p\in[0,1]\subseteq\mathbb{R}$ can associated to the rarity: $k$ kays as follows:

$p:\eq 10^{-k}$ and
k:=−log10(p)
, or k:=−ln(p)ln(10)
(see log10 (function) and ln (function) for more information)
- Be aware of the warning in the note below when using $\log$ , $\lg$ ect in software^{[Note 1]}

Properties

Suppose an event $\mathrm{X}$ occurs with rarity $u$ kays, and independently an event $\mathrm{Y}$ occurs with rarity $v$ kays then:

The event of both $\mathrm{X}$ and $\mathrm{Y}$ occurring has rarity $u+v$

If $p$ is the probability corresponding to $u$ kays and $q$ the probability corresponding to $v$ kays then:

$u+v$ kays corresponds to the probability $pq$

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

Table

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

Jump up ↑ Be aware that seldom some systems use $\log$ to refer to natural log, more commonly written $\ln$ ; such systems will use $\text{lg}$ for $\log_{10}$ instead. Because of the problems this caused most "recent" systems will use $\log_{10}$ and $\ln$ , consider $\log$ by itself ambiguous, $\text{lg}$ was unfortunately used as a shorthand for what we'd now write $\log_{2}$ 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 $0$ to $9$ (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 $\log_{10}$

[1] Jump up ↑ Be aware that seldom some systems use $\log$ to refer to natural log, more commonly written $\ln$ ; such systems will use $\text{lg}$ for $\log_{10}$ instead. Because of the problems this caused most "recent" systems will use $\log_{10}$ and $\ln$ , consider $\log$ by itself ambiguous, $\text{lg}$ was unfortunately used as a shorthand for what we'd now write $\log_{2}$ 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 $0$ to $9$ (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 $\log_{10}$

[Note 1]

Kay (unit)

Contents

Definition

Formula

Properties

Table

Notes

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