# Log5

Log 5 is a formula invented by Bill James 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.

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

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

A few notable properties exist:

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

It may also be conveniently rewritten using the odds ratio as ${\frac {p_{A,B}}{q_{A,B}}}={\frac {p_{A}}{q_{A}}}\times {\frac {q_{B}}{p_{B}}}.$ Here as before $q_{A,B}=1-p_{A,B}$ .