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, $p\in[0,1]\subseteq\mathbb{R}$ the corresponding values are:

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

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

• $$p:\eq 10^{-k}$$

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

• $$p:\eq 1-10^{-k}$$

Selected examples

value, $v$	$v\ \mathbf{k}$ (rarity)		$v\ \mathbf{k}'$ (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
$\vdots$	$\vdots$
$\infty$	0 (impossibility)		1 (certainty)