Fenway

06-05-2010, 02:02 PM

BOSOX@APPLE.EASE.LSOFT.COM

A warning now, hit "next" if you aren't interested in abstruse

mathematics laden with simplifying assumptions. :) Just thought I'd

lay out a few formulas that might be useful as a "first order"

approximation.

I was looking for a framework from which to estimate the playoff odds

in the simplest possible situation: a race for a single playoff spot

between two teams of comparable quality without head-to-head games to

confuse the situation. Assume that the performance of each team in

each game is independently binomially distributed.

What are the odds of a team that is trailing by X games making up the

difference if there are N games remaining in the season?

The variance of a binomial distribution is calculated by n * p * (1-p)

where n is the number of trials/games and p is the probability of

success/winning. Because the product p * (1-p) varies only slightly

in the range we are discussing (teams between .500 and .600), we can

approximate this as being n * .5 * .5 = n/4. Thus the variance in the

Red Sox final record is approximately 106/4 = 26.5.

When calculating the sum or difference of variables (i.e. comparing

two teams over the remainder of the season) the variances add. Thus

the variance in the number of games the Rays win compared to the

number of games the Red Sox win is roughly 26.5 + 26.75 = 53. The

standard deviation is the square root of the variance, or ~7.3 games

in the standings.

The Rays are presently 4.5 games ahead of the Red Sox. Thus the Red

Sox only catch the Rays if they are 4.5 games better over the

remainder of the season. Now I'm assuming that the Red Sox and Rays

are of equal quality, thus that would be 4.5 games better than the

expected mean. Dividing by the standard deviation, we see this is

+0.6 standard deviations.

Looking that up in a Z-score table, we see that the Red Sox would fall

short 72.6% of the time while catching the Rays 27.4% of the time.

Some thoughts...

* The head-to-head games clearly break the assumption of independence.

A swing in THOSE games in effect counts twice.

* The Red Sox were thought to be the better team in March. If their

expectation were two games better over the remainder of the season,

then they would only need to be 2.5 games above expectation to catch

the Rays. That would improve the Red Sox chances of catching the Rays

to 37%.

* Because this depends on the square root of the variance, the

standard deviation will shrink only slowly in the coming months. With

64 games left to play (last week of July?), the variance for a single

team would be 16 games and for a pair of teams compared would be 32

games. That would shrink the standard deviation from its present 7.3

games to "just" 5.7 games. With one month left of play, the standard

deviation when comparing two teams is still 4 games. This is why

old-time baseball fans will continually remind you, "there's a lot of

baseball left".

In conclusion, I'm liking the Red Sox chances much better these days.

It still won't come as a surprise to me if they fall short, but they

clearly have a realistic shot at the playoffs at this point.

-Ted

A warning now, hit "next" if you aren't interested in abstruse

mathematics laden with simplifying assumptions. :) Just thought I'd

lay out a few formulas that might be useful as a "first order"

approximation.

I was looking for a framework from which to estimate the playoff odds

in the simplest possible situation: a race for a single playoff spot

between two teams of comparable quality without head-to-head games to

confuse the situation. Assume that the performance of each team in

each game is independently binomially distributed.

What are the odds of a team that is trailing by X games making up the

difference if there are N games remaining in the season?

The variance of a binomial distribution is calculated by n * p * (1-p)

where n is the number of trials/games and p is the probability of

success/winning. Because the product p * (1-p) varies only slightly

in the range we are discussing (teams between .500 and .600), we can

approximate this as being n * .5 * .5 = n/4. Thus the variance in the

Red Sox final record is approximately 106/4 = 26.5.

When calculating the sum or difference of variables (i.e. comparing

two teams over the remainder of the season) the variances add. Thus

the variance in the number of games the Rays win compared to the

number of games the Red Sox win is roughly 26.5 + 26.75 = 53. The

standard deviation is the square root of the variance, or ~7.3 games

in the standings.

The Rays are presently 4.5 games ahead of the Red Sox. Thus the Red

Sox only catch the Rays if they are 4.5 games better over the

remainder of the season. Now I'm assuming that the Red Sox and Rays

are of equal quality, thus that would be 4.5 games better than the

expected mean. Dividing by the standard deviation, we see this is

+0.6 standard deviations.

Looking that up in a Z-score table, we see that the Red Sox would fall

short 72.6% of the time while catching the Rays 27.4% of the time.

Some thoughts...

* The head-to-head games clearly break the assumption of independence.

A swing in THOSE games in effect counts twice.

* The Red Sox were thought to be the better team in March. If their

expectation were two games better over the remainder of the season,

then they would only need to be 2.5 games above expectation to catch

the Rays. That would improve the Red Sox chances of catching the Rays

to 37%.

* Because this depends on the square root of the variance, the

standard deviation will shrink only slowly in the coming months. With

64 games left to play (last week of July?), the variance for a single

team would be 16 games and for a pair of teams compared would be 32

games. That would shrink the standard deviation from its present 7.3

games to "just" 5.7 games. With one month left of play, the standard

deviation when comparing two teams is still 4 games. This is why

old-time baseball fans will continually remind you, "there's a lot of

baseball left".

In conclusion, I'm liking the Red Sox chances much better these days.

It still won't come as a surprise to me if they fall short, but they

clearly have a realistic shot at the playoffs at this point.

-Ted