Thursday, March 29, 2018


Hockey and basketball are better than football in one important regard: There are enough games played to develop a fairly robust Bradley-Terry probability model, and the main virtue of this model is that it allows us to get a reasonable idea of how likely one team is to beat another. So let's jump in and see what the numbers predict.

In college hockey, the Bradley-Terry rating is called KRACH (Ken's Rankings for American College Hockey), and I've pulled the latest numbers from College Hockey News.

Team KRACH Prob. to win
Prob. to win
Notre Dame 468 66.64% 40.00%
Michigan 234.3 33.36% 14.39%
Ohio State 370.2 61.25% 30.54%
Minnesota-Duluth 234.2 38.75% 15.07%

The bad news: According to KRACH, Michigan has the worst chance of any team to win the Frozen Four. This is because they face the hardest path: KRACh thinks Notre Dame is significantly better than Ohio State, and Michigan is worse than both of them. If Michigan made it to the final and found UMD there, they would be a slight favorite on truly neutral ice.

The good news: Hockey Plinko is still Hockey Plinko. A hot goalie can steal a game, and a single weird bounce can make an enormous impact. Meanwhile, nobody has been playing better hockey than Michigan recently.

Bradley-Terry rankings for college basketball (and football can be found at dbaker's site. You can find either ratings that care about margin of victory or ones that don't.

Basic Bradley-Terry

Team Rating Prob. to win
Prob. to win
Villanova 21.619 61.38% 41.36%
Kansas 13.600 38.62% 21.83%
Michigan 11.483 55.24% 21.53%
Loyola-Chicago 9.3029 44.76% 15.29%

Margin-Aware Bradley-Terry

Team Rating Prob. to win
Prob. to win
Villanova 11.277 68.20% 46.87%
Kansas 5.2574 32.80% 16.13%
Michigan 5.9095 58.80% 23.69%
Loyola-Chicago 4.1399 41.20% 13.32%

Both sets of rankings give the basketball team a better chance of winning a title than the hockey team. Bare bones, who-did-you-beat Bradley-Terry gives Michigan a 21.53% chance of winning it all. If you add in margin of victory, Michigan squeaks up to a 23.59% chance of winning.

The rankings also show how good Loyola's resume is looking these days. Michigan is somewhere between a 55.24% and 58.80% favorite in Saturday's game. That's not much! Villanova, meanwhile is either a 61.38% or 68.20% favorite against Kansas. People would rightly view it as an upset if Kansas wins, but it wouldn't have the shock value of a commuter school beating a name brand team, even though the numbers say Michigan is closer to Loyola than Kansas is to Villanova.

To get into the numbers a little deeper, both sets of numbers give Michigan about a 34.5% chance of beating Villanova in a final. They disagree about whether Michigan is a favorite or a dog to Kansas. the basic numbers have them at 45.78%, while the margin-aware numbers are at 52.92%, so that sounds like an even battle.

Finally, the big questions: What are the chances we win everything that's up for grabs? What are the chances that we win at least one title? Well, the chance that we win either of two independent events is the sum of their probabilities. Therefore, depending on which basketball rating you use, we have either a 35.92% or 38.07% chance of one team bringing home a title. And to compute the probability that both happen, you multiply them together, giving us either a 3.10% or 3.41% chance of bringing home a truly ridiculous amount of hardware. So I'm saying there's a chance.


Unknown said...

Actually the chance to win either is a+b-a*b or about %3 less than you calculated.

Scott said...

Your probability calculations are off: the probability that one of two independent events occurs is not the sum of their probabilites; if it were, then if you flipped two coins simultaneously, this would imply a 100% chance that one of them would come up Heads.

Using 14.39% for hockey and 23.69% for basketball, here are the probabilities:

Both win: 3.4%
At least one wins: 34.7%
Exactly one wins: 31.2%
Neither wins: 65.3%