statistical
Log5
Also known as: Log5 Formula, Bill James Log5
Log5 is a formula that combines two independent rate statistics (typically batter vs. pitcher) into a single matchup-adjusted probability, accounting for the league average. It is the workhorse equation for head-to-head probability modeling in baseball.
Log5 solves a specific problem: given a batter who hits at rate A against league average and a pitcher who allows at rate B against league average, what is the expected rate when they face each other? Naively averaging A and B gives the wrong answer because it ignores the baseline. Log5 corrects for it.
The formula: P(event) = (A × B / L) / (A × B / L + (1 - A) × (1 - B) / (1 - L)), where A is the batter’s rate, B is the pitcher’s rate against, and L is the league rate. It generalizes anywhere two independent agents interact against a common baseline.
Worked example. A batter has a true home run rate per plate appearance of 5% (league average 3.5%). The starting pitcher has allowed home runs at a rate of 4% per PA (league average 3.5%). Log5 HR rate = (0.05 × 0.04 / 0.035) / (0.05 × 0.04 / 0.035 + 0.95 × 0.96 / 0.965) = 0.057 / (0.057 + 0.945) = 0.0568, or about 5.7% per PA. Across four expected plate appearances, the batter has a (1 - 0.943^4) = 21.4% chance of hitting at least one home run, which can be compared directly against the home run prop line.
Elite Picks’ MLB model uses Log5 (and Log5-adjusted variants for park factor and platoon splits) on every batter-vs-pitcher matchup. The output feeds into the home run, total bases, and hits-allowed probability distributions. Log5 is one of the few statistical formulas that holds up across decades of baseball analysis with minimal modification.
Example
A batter with a .330 OBP faces a pitcher who allows a .290 OBP, in a league averaging .320. Log5 OBP = (0.330 × 0.290 / 0.320) / ((0.330 × 0.290 / 0.320) + ((1 - 0.330) × (1 - 0.290) / (1 - 0.320))) = 0.300.