k and k' values

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Worked on this before bed - needs finishing! Alec (talk) 23:14, 7 February 2018 (UTC)

Definition

Given a probability, [ilmath]p\in[0,1]\subseteq\mathbb{R} [/ilmath] the corresponding values are:

• [math]k:\eq\frac{-\ln(p) }{\ln(10) } [/math], higher values indicate the event we have the probability for is rarer, eg [ilmath]k\eq 6[/ilmath] is 1 in 1,000,000 (1 million), or [ilmath]p\eq 0.000001[/ilmath]
• [math]k':\eq\frac{-\ln(1-p)}{\ln(10)} [/math], higher values indicate the event we have the probability for is more common, eg [ilmath]k'\eq 6[/ilmath] is 999,999 in 1,000,000, or [ilmath]p\eq 0.999999[/ilmath]

Given a k-value, [ilmath]k\in\mathbb{N}_{\ge 0} [/ilmath] then the corresponding probability is:

• [math]p:\eq 10^{-k} [/math]

Given a k'-value, [ilmath]k'\in\mathbb{N}_{\ge 0} [/ilmath] then the corresponding probability is:

• [math]p:\eq 1-10^{-k} [/math]

Selected examples

value, [ilmath]v[/ilmath]	[ilmath]v\ \mathbf{k} [/ilmath] (rarity)		[ilmath]v\ \mathbf{k}' [/ilmath] (commonality)
	(As probability)
0	1 (certainty)		0 (impossibility)
1	0.1	1 in 10	0.9	9 in 10
2	0.01	1 in 100	0.99	99 in 100
3	0.001	1 in 1,000	0.999	999 in 1,000
4	0.0001	1 in 10,000	0.9999	9,999 in 10,000
[ilmath]\vdots[/ilmath]	[ilmath]\vdots[/ilmath]
[ilmath]\infty[/ilmath]	0 (impossibility)		1 (certainty)