# 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: [ilmath]\infty[/ilmath] 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, [ilmath]p\in[0,1]\subseteq\mathbb{R} [/ilmath] can associated to the rarity: [ilmath]k[/ilmath] kays as follows:

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

## Properties

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

• The event of both [ilmath]\mathrm{X} [/ilmath] and [ilmath]\mathrm{Y} [/ilmath] occurring has rarity [ilmath]u+v[/ilmath]

If [ilmath]p[/ilmath] is the probability corresponding to [ilmath]u[/ilmath] kays and [ilmath]q[/ilmath] the probability corresponding to [ilmath]v[/ilmath] kays then:

• [ilmath]u+v[/ilmath] kays corresponds to the probability [ilmath]pq[/ilmath]

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

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