Math thread

[billhoward]

My high school physics teacher was also terrible.  He quit teaching a couple years after I graduated to become a priest :-O

Now the boys in the parish will be nervous about being around physics, too.

[French Rage]

Any chance we could get a Krach graph of college football for this year?

[jkahn]

Any chance we could get a Krach graph of college football for this year?

Not in graphic form but:
http://mattcarberry.com/ZRatings/Z-CFB.HTM
http://www.vaporia.com/sports/collegefootballkrach.html
It's interesting how the two versions differ as they use different weightings for the dummy game necessary to avoid having teams with an infinite rating.

[Robb]

Any chance we could get a Krach graph of college football for this year?

Not in graphic form but:
http://mattcarberry.com/ZRatings/Z-CFB.HTM
http://www.vaporia.com/sports/collegefootballkrach.html
It's interesting how the two versions differ as they use different weightings for the dummy game necessary to avoid having teams with an infinite rating.

It's also a little insane how much better Auburn's rating is than everyone else (in both rankings).  Matt makes Auburn a 6:1 favorite over Stanford, while Vaporia has them at a whopping 49-1.  I'd take either of those bets straight up!

[jkahn]

Any chance we could get a Krach graph of college football for this year?

Not in graphic form but:
http://mattcarberry.com/ZRatings/Z-CFB.HTM
http://www.vaporia.com/sports/collegefootballkrach.html
It's interesting how the two versions differ as they use different weightings for the dummy game necessary to avoid having teams with an infinite rating.

It's also a little insane how much better Auburn's rating is than everyone else (in both rankings).  Matt makes Auburn a 6:1 favorite over Stanford, while Vaporia has them at a whopping 49-1.  I'd take either of those bets straight up!

Actually, Matt has Oregon ahead of Auburn, and they'd be 49-1 vs. Stanford.  Vaporia (John Wobus) has Auburn at #1 and 6:1 vs. Stanford.  While I think these are the the best way to order ranking based on results, the small sample size plays havoc with the ratios.

[Trotsky]

Interesting that Carberry has the Big 12 as the best conference despite not having a member in the top 8.  I guess it's like golf -- have the best worst hole.

[kingpin248]

Interesting that Carberry has the Big 12 as the best conference despite not having a member in the top 8. I guess it's like golf -- have the best worst hole.

the small sample size plays havoc with the ratios

Greg is right here. The conference rating is simply the average of the non-conference RRWPs of the conference's teams (listed under 'nCWP' in my tables). The Big 12 has eight members in the top 25 and all twelve are in the upper half of I-A. Three SEC teams are rated worse than the lowest Big 12 team, Kansas. The Jayhawks' SOS is also the lowest in the league, at 38.

This highlights another problem with the application of KRACH to college football - the "Appalachian State problem." Kansas, in particular, would likely be ranked singificantly lower if its season-opening loss to North Dakota State were accounted for. Virginia Tech is now at #20, but they were in the top ten earlier in the season, because the James Madison loss didn't impact them. The Colley Matrix (a BC$ component) uses a novel method to account for FCS teams - grouping the FCS teams until they "look" like FBS teams.

Actually, Matt has Oregon ahead of Auburn, and they'd be 49-1 vs. Stanford.

The top two flip-flopped after Saturday's games, and it had nothing to do with what either of them did - it was caused by LSU's loss to Arkansas. That dropped LSU into the "main field" of teams.

Same ratings as linked above, sorted by conference.

The biggest thing to note is that a team currently ranked #71 (#66 in the Wobus KRACH) is one win away from an $18 million payout.

[nr53]

I've wondered for the past few years what the NHL standings would look like using KRACH but I don't know how to set that up myself. Anyone care to point me towards a link that could explain the process to me?

[jkahn]

Interesting that Carberry has the Big 12 as the best conference despite not having a member in the top 8. I guess it's like golf -- have the best worst hole.

the small sample size plays havoc with the ratios

Greg is right here. The conference rating is simply the average of the non-conference RRWPs of the conference's teams (listed under 'nCWP' in my tables).

I don't see any relationship between my comment and Greg's comment here.  My comment on the ratios and small sample sizes was that given the small number of games on the college football schedule, a ratio such as 49:1 Oregon vs. Stanford certainly doesn't meet our realistic expectation of the true odds, even though it's an appropriate way to establish a ranking order of the teams.  It's also strongly influenced by having on of those teams being undefeated.  With a 30 game college hockey schedule, we see the ratios getting much closer to our sense of reality.

[kingpin248]

I don't see any relationship between my comment and Greg's comment here.

True. My bad.

I've wondered for the past few years what the NHL standings would look like using KRACH

I do the pro leagues too! ::crazy:: 2010-11 NHL.
The records given in this table don't match the NHL standings; they are W-L-T, and do not include shootout results or account for the overtime loser point.

[jkahn]

Being a part-time math geek when I'm not being a hockey geek, I was asked by email today the following question:
What are the odds of the Packers and Bears 1) both being in the playoffs and 2) playing each other in the playoffs and 3) playing each other in the NFC championship game, assuming each NFL team has a 50% chance of winning each game?

While I quickly responded with the following to the fellow who asked, I thought the question was thought provoking enough that readers of this thread might enjoy a shot at coming up with a more complete answer.

My quick reply:
"First, the easy part.
Let's figure the odds of both the Packers and Bears being in the playoffs.
Since there a 4 teams in the NFC North, and the division champ automatically makes the playoffs, there's a 50% that one of the Bears and Packers will be division champs.
Then there are two spots left for wildcards and 12 non-division champs.  So if the Bears or Pack are champs, the non-champ has a 2/12 chance of making the playoffs.  So the odds of both making the playoffs, one as champ and the other as wildcard, is 1/2 times 2/12 = 1/12.
Now actually it is slightly higher than that, because even if the Lions or Vikes finish first, it is remotely possible that the Bears and Pack could be the two wildcards, but that's unlikely and requires more math than I have time to spend - since it's not just a question of each team having an independent chance - if one team from a division gets the first wildcard, it's less likely that another team in that division will get the second, because that team needed an above average record, which means its wins may be losses for other teams in the division.
The same issue with dependent variables clouds the next part of the analysis.  If the Bears at division champ had an equal chance to be seeded anywhere between 1 and 4, and the Pack as wildcard had an equal chance to be either 5 or 6, we could do the analysis.  For instance, then 25% of the time the Bears would be #3 and 50% of the time the Pack would be #6, giving them a 1/4 times 1/2 chance or 1/8 of being seeded 3-6 and meeting in the first round.  You would then multiply that by the 1/12, and there's be a 1/96 chance that one would be seeded #3 and one #6 and meet in the first round.  Similarly, a 1/96 chance of being #4 and #5, so a total of 2/96, or 1/48 of meeting in the first round.  However, that analysis is faulty, because it assumes that the records of the Pack and Bears are independent variables.  However, we logically know that it is much less likely that two teams from the same division will be the 4 and 5 seeds than the odds of them being #3 and 6.  That's because the 4th seed is the champ with the weakest record and it is likely the top wildcard will have a better record and not be from that division.  So the analysis gets very hairy at this point.
Perhaps more later if I have time."

[Trotsky]

What are the odds of the Packers and Bears 1) both being in the playoffs and 2) playing each other in the playoffs and 3) playing each other in the NFC championship game, assuming each NFL team has a 50% chance of winning each game?

The odds as of today are 1.

[KeithK]

Lets ask a more challenging question: what are the odds of Bengals and the Browns meeting in the AFC Championship game?  Somehow I think there's a better chance that people can sense the future through psychic powers...

(Implicitly rejecting the coin flip assumption.)

[Trotsky]

Super Bowl Squares odds.  (warning: url is for a gambling site that might be flagged by some work browsers so follow at own risk)

Here is a table with the results from almost 20 years of NFL games.  I assume since this is a neutral site game one is better off taking the average of each {x,y} and {y,x} pair.  The precedence of numbers (7, 0, 4, 3...) is not that surprising.

[Trotsky]

Nerd crack.