Log5 Equation / Series Probabilities

[rluzinski]

Ever wonder how often a .550 team beats a .450 team? I did, and after a little searching I stumbled onto the Log5 equation. Introduced by Bill James in his 1981 Bill James Baseball Abstract, it states:

 A - A * B WPct = ----------------- A + B - 2 * A * B 

Where:

 

A = Team A's Winning %

B = Team B's Winning %

 

WPct tells us what % of games team A will beat team B, on average.

 

What if we want to calculate the odds of winning 1 game in a 2 game series, however? The first step is to calculate the odds of winning 1 game "one way". The odds of winning the first and losing the second:

 

PROB = WPct * (1 - Wpct)

 

WPct is the probabilty of winning the first game and 1- WPct is the probabilty of losing the second game. The probability of BOTH occuring is simply the product. But that's only one way to win 1 game of a 2 game series, however. The team could lose the first and win the second and still win 1 game in the series. Common sense tells that that the probability of winning a specific number of games in a series is then:

 

PROB(one way) * number of ways

 

So, the probabilty of winning 1 game in a two game series is:

 

WPct * (1-WPct) * 2

 

If we want to calculate the probability of winning any number of games in any length of a series we need to create a more flexible equation. We'll define the two variables we need as:

 

G = total games in the series

W = number of wins in a series

 

The probabilty of winning "one way" is then simply:

 

[WPct ^ W] * [(1-WPt) ^ (G-W)]

 

All that's left to figure out is the "number of ways" part. We could figure it by hand if we wanted to. For a 3 game series, the number of ways to win 1 game is:

 

WLL

LWL

LLW

 

Three ways to win 1 game in a 3 game series. It just so happens that the Combination function can calculate just how many ways you can win any number of games in any length series:

 

"number of ways" = C(G, W)

 

Putting the final equation together:

 

PROB(G,W) = [WPct ^ W] * [(1-WPt) ^ (G-W)] * C(G, W)

 

A few more points to finish this post. Since the home team wins 4% more games, simply add .04 to their WPct. Also, I use the pythagorean winning percentages for the Log5 formula since I feel it predicts future success better.

 

If anyone is interested in the actual computer code I will be happy to type it in. Let me know if there's any questions or suggestions!

 

Links:

 

Estimating one-game winning percentages

 

Combination Formula

 

Pythagorean Theorum

[Oswald Chesterfield Cobble]

is there anyway way to account for a team being at home or on the road?

[rluzinski]

is there anyway way to account for a team being at home or on the road?

 

The average home team wins at about a .540 clip. So:

 

WPct(home team) = WPct + 0.040

 

I was going to add that when I finish the post http://forum.brewerfan.net/images/smilies/smile.gif

[And That]

I await in unguarded anticipation for the continuation of the Log5 equation into predicting a series of games. Don't leave us hanging, Russ, we need your formulas.

[rluzinski]

Awwww nuts... I forgot, to be honest. I figured it out and put it on my calculator.... now I have to remember where those equations came from!

[rluzinski]

I'm going to start calculating series probabilites again, with a little twist. I'll be using raw stats to generate expected runs scored & runs against (using the BaseRuns run estimator). I'll then use the pythagorean to calculate expected record, and throw those winning % in the Log5 equation. As before, I'll give the home team a 4% bump.