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'''Log 5''' is a formula invented by [[Bill James]]<ref>{{cite news|title=Chancesis: The Origins of Log5|url=http://www.chancesis.com/2010/10/03/the-origins-of-log5/|accessdate=2013-03-07}}</ref> to estimate the probability that team A will win a game, based on the true winning percentage of Team A and Team B. It's equivalent to the [[Bradley-Terry-Luce]] model used for [[paired comparisons]], the [[Elo rating system]] used in chess and the [[Rasch model]] used in the analysis of categorical data.<ref name="ASLog5">{{cite news|title=Baseball, Chess, Psychology and Pychometrics: Everyone Uses the Same Damn Rating System|url=http://angrystatistician.blogspot.com/2013/03/baseball-chess-psychology-and.html|accessdate=2013-12-29}}</ref>

Let <math>p_i</math> be the fraction of games won by team <math>i</math> and also let <math>q_i = 1-p_i</math> be the fraction of games lost by team <math>i</math>. The Log5 estimate for the probability of A defeating B is <math>p_{A,B} = \frac{p_A-p_A\times p_B}{p_A+p_B-2\times p_A\times p_B}.</math>

A few notable properties
*If <math>p_A = 1</math>, Log5 will always give A a 100% chance of victory
*If <math>p_A = 0</math>, Log5 will always give A a 0% chance of victory
*If <math>p_A = p_B</math>, Log5 will always return a 50% chance of victory for either side
*If <math>p_A = 1/2</math>, Log5 will give A a <math>1-p_B</math> probability of victory.

It may also be conveniently rewritten using the [[odds ratio]]<ref name="ASLog5" /> as <math>\frac{p_{A,B}}{q_{A,B}} = \frac{p_A}{q_A}\times \frac{q_B}{p_B}.</math>

Here as before <math>q_{A,B}=1-p_{A,B}</math>.

==References==
{{Reflist}}

[[Category:Bill James]]
[[Category:Baseball terminology]]
[[Category:Baseball statistics]]
[[Category:Sports records and statistics]]