The Mathematics of Beating OSU

Another week, another titanic shift in the Pairwise rankings. As far as we're concerned, the big news is Michigan jumping up to #2 after Notre Dame's split with Ohio State. The Wolverines are tied with Northeastern for that spot, but own the comparison with the Huskies, and thus the tiebreaker. It really is a knife's edge, though. Take a look at the full chart at CHN: Michigan, Northeaster, Vermont, and Notre Dame all have 21 or 22 comparison wins over the field. Lose a game, and suddenly you're the #5 overall seed. And sometimes you don't even have to lose to have a comparison flip on you. Remember the TUC cliff?

As a refresher, the Pairwise Rankings consider four factors when comparing two teams: head-to-head record, RPI, record against common opponents, record against teams under consideration (TUCs), but you have to have played 10 TUCs for that comparison to be used.

For instance, let's consider Michigan and Miami. Head-to-head record is 2-2, so two points to each team for that. Michigan's RPI is better, so they get a point. Miami's record against common opponents is better: point, Redhawks. Miami has only played 8 TUCs, so that comparison is not used. Michigan and Miami are tied 3-3 here, and we use RPI to break the tie. Michigan takes the comparison. HOWEVA. Miami has played 8 TUCs, which are defined as the top 25 teams by RPI. They play Ohio State twice to close out the season. As OSU is currently a TUC, that would allow them to meet the 10-game requirement. If both teams win out, Miami's record against common opponents is still better and now their record against TUCs is better, flipping the comparison over to them. Maddening, I know, but that's the way the Pairwise works.

What that means is that this weekend's series against the Buckeyes is much bigger than even a regular UM/OSU brawl: It's fighting for position at the top of the Pairwise, fighting for a spot in the Grand Rapids regional against the soft underbelly of Atlantic Hockey or College Hockey America. Win, get a Miami loss, and life is looking a lot easier. If not, drop below Notre Dame and you have to fight harder to not get sent to Minneapolis.

So, what are our chances? KRACH is here to answer that. KRACH is meant to use ratios between teams' KRACH ratings to give a predicted winning percentage. For example, the expected winning percentage of a team with a KRACH of 150 playing a team with a KRACH of 50 is .750.

Michigan's expected win pct. vs. OSU = Michigan KRACH / (OSU KRACH + Michigan KRACH)

Michigan currently sports a shiny KRACH rating of 402, while OSU is up to a rather healthy 197.8. That gives Michigan an expected winning percentage of .6702. The probability of two independent events both occurring is the product of the probability of either one occurring separately (i.e., P(A and B) = P(A)*P(B)). Michigan has a 67.02% chance of winning on either night against OSU, so the probability of a sweep is 44.92%, going by KRACH. Not great odds, but I'll take 'em. OSU only has a 10.88% chance of a sweep, leaving a 44.2% chance that we split.*

Michigan's been playing decent hockey since the Bowling Green loss, while OSU has been doing weird things, like only getting a pair of ties out of four opportunities against Western and State (though they got a win againt Western in the shootout) while beating Notre Dame and almost sweeping them but for some late-period heroics by the Irish.

Meanwhile, root for Northern to upset Miami down in Oxford. We'll take any help we can get. And really pull for the Bucks when Miami comes calling.

*Odds of Michigan winning out for the rest of the season? A moderately robust 30.1%.

UPDATE: Excellent catch/clarification by Brian, reproduced here verbatim: "Miami's TUC record is actually 6-4-2, but head to head games are excluded from that category in each individual comparison, leaving Miami at 4-2-2 for the purposes of its Michigan comparison." So for the purposes of everyone not in the CCHA, Miami is a TUC.