# Log5

Log 5 is a formula invented by Bill James[1] to estimate the probability that team A will win a game, based on the true winning percentage of Team A and Team B.

It is 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.[2]

Let ${\displaystyle p_{i}}$ be the fraction of games won by team ${\displaystyle i}$ and also let ${\displaystyle q_{i}=1-p_{i}}$ be the fraction of games lost by team ${\displaystyle i}$.

The Log5 estimate for the probability of A defeating B is ${\displaystyle p_{A,B}={\frac {p_{A}-p_{A}\times p_{B}}{p_{A}+p_{B}-2\times p_{A}\times p_{B}}}}$.

A few notable properties exist:

• If ${\displaystyle p_{A}=1}$, Log5 will always give A a 100% chance of victory.
• If ${\displaystyle p_{A}=0}$, Log5 will always give A a 0% chance of victory.
• If ${\displaystyle p_{A}=p_{B}}$, Log5 will always return a 50% chance of victory for either team.
• If ${\displaystyle p_{A}=1/2}$, Log5 will give A a ${\displaystyle 1-p_{B}}$ probability of victory.

It may also be conveniently rewritten using the odds ratio[2] as ${\displaystyle {\frac {p_{A,B}}{q_{A,B}}}={\frac {p_{A}}{q_{A}}}\times {\frac {q_{B}}{p_{B}}}.}$

Here as before ${\displaystyle q_{A,B}=1-p_{A,B}}$.

## References

1. ^ "Chancesis: The Origins of Log5". Archived from the original on April 12, 2012. Retrieved 2013-03-07.
2. ^ a b "Baseball, Chess, Psychology and Pychometrics: Everyone Uses the Same Damn Rating System". Retrieved 2013-12-29.