# Paths To Victory: A Purely Hypothetical Construct

I write this in the full realization that it is based on a series of assumptions that could, at any moment, render this diary moot, but I thought I might share it all the same.

Much has been said in some articles recently, as well as some threads here, about Michigan’s chances of making it to the Championship Game, not to mention winning the NCAA Tournament. It dawned on me that we can estimate these chances for each of the remaining possible paths to the game, and indeed, to the title.

First, of course, we would need to beat Kansas, and at least at the time of this writing and the time that I ran the simulated matchup, Michigan was a slight favorite over at Massey – 53%. So, for the sake of argument, assume that we beat Kansas in this scenario:

MICHIGAN VS. KANSAS |
ELITE EIGHT |
FINAL FOUR |
EST. PROB. |

0.53 |
0.84 |
0.72 |
32.054% |

0.53 |
0.84 |
0.61 |
27.157% |

0.53 |
0.84 |
0.54 |
24.041% |

0.53 |
0.84 |
0.43 |
19.144% |

0.53 |
0.43 |
0.72 |
16.409% |

0.53 |
0.43 |
0.61 |
13.902% |

0.53 |
0.43 |
0.54 |
12.307% |

0.53 |
0.43 |
0.43 |
9.800% |

The table above outlines possible paths to the Championship game. If we beat Kansas, then it is either Florida (43% estimated probability) or Florida Gulf Coast (84% estimated probability). If we were to pass this hurdle in either scenario, it would lead to a game against either Marquette (72%), Syracuse (61%), Miami (YTM) (54%) or Indiana (44%).

So, you can see here the estimated probabilities for each of those paths. Obviously, the path which would have us beating Florida Gulf Coast and Marquette turns out to be the easiest and most attractive, and the Kansas-Florida-Indiana route would be the most unlikely and harrowing. However, as you’ll note, there are no non-zero chances.

Of course, there are eight potential matchups coming from the other two regions at this point, at least for the time being. Here’s where I make an admittedly unscientific jump for purposes of simplicity – I did not actually calculate the independent paths for each of the teams to reach the championship game. Instead, I rather went with the “Say, for the moment, this happens and they reach the final game…” approach in the full understanding that, for seven of those teams, it won’t happen. I sort of address this in a moment.

So, we have Louisville, Oregon, Michigan St., Duke, Wichita St. La Salle, Arizona and Ohio State still sitting out there. This is a purely hypothetical scenario where you knew who the opponent would be in advance essentially.

LOUISVILLE |
|||||

MICHIGAN VS. KANSAS |
ELITE EIGHT |
FINAL FOUR |
EST. PROB TO REACH NC GAME |
EST. PROB TO BEAT LOUISVILLE |
CHANCE OF WINNING ON PATH |

0.53 |
0.84 |
0.72 |
32.054% |
0.31 |
9.937% |

0.53 |
0.84 |
0.61 |
27.157% |
0.31 |
8.419% |

0.53 |
0.84 |
0.54 |
24.041% |
0.31 |
7.453% |

0.53 |
0.84 |
0.43 |
19.144% |
0.31 |
5.935% |

0.53 |
0.43 |
0.72 |
16.409% |
0.31 |
5.087% |

0.53 |
0.43 |
0.61 |
13.902% |
0.31 |
4.310% |

0.53 |
0.43 |
0.54 |
12.307% |
0.31 |
3.815% |

0.53 |
0.43 |
0.43 |
9.800% |
0.31 |
3.038% |

OREGON |
|||||

MICHIGAN VS. KANSAS |
ELITE EIGHT |
FINAL FOUR |
EST. PROB TO REACH NC GAME |
EST. PROB TO BEAT OREGON |
CHANCE OF WINNING ON PATH |

0.53 |
0.84 |
0.72 |
32.054% |
0.64 |
20.515% |

0.53 |
0.84 |
0.61 |
27.157% |
0.64 |
17.381% |

0.53 |
0.84 |
0.54 |
24.041% |
0.64 |
15.386% |

0.53 |
0.84 |
0.43 |
19.144% |
0.64 |
12.252% |

0.53 |
0.43 |
0.72 |
16.409% |
0.64 |
10.502% |

0.53 |
0.43 |
0.61 |
13.902% |
0.64 |
8.897% |

0.53 |
0.43 |
0.54 |
12.307% |
0.64 |
7.876% |

0.53 |
0.43 |
0.43 |
9.800% |
0.64 |
6.272% |

MICHIGAN STATE |
|||||

MICHIGAN VS. KANSAS |
ELITE EIGHT |
FINAL FOUR |
EST. PROB TO REACH NC GAME |
EST. PROB TO BEAT MICHIGAN STATE |
CHANCE OF WINNING ON PATH |

0.53 |
0.84 |
0.72 |
32.054% |
0.49 |
15.707% |

0.53 |
0.84 |
0.61 |
27.157% |
0.49 |
13.307% |

0.53 |
0.84 |
0.54 |
24.041% |
0.49 |
11.780% |

0.53 |
0.84 |
0.43 |
19.144% |
0.49 |
9.380% |

0.53 |
0.43 |
0.72 |
16.409% |
0.49 |
8.040% |

0.53 |
0.43 |
0.61 |
13.902% |
0.49 |
6.812% |

0.53 |
0.43 |
0.54 |
12.307% |
0.49 |
6.030% |

0.53 |
0.43 |
0.43 |
9.800% |
0.49 |
4.802% |

DUKE |
|||||

MICHIGAN VS. KANSAS |
ELITE EIGHT |
FINAL FOUR |
EST. PROB TO REACH NC GAME |
EST. PROB TO BEAT DUKE |
CHANCE OF WINNING ON PATH |

0.53 |
0.84 |
0.72 |
32.054% |
0.48 |
15.386% |

0.53 |
0.84 |
0.61 |
27.157% |
0.48 |
13.035% |

0.53 |
0.84 |
0.54 |
24.041% |
0.48 |
11.540% |

0.53 |
0.84 |
0.43 |
19.144% |
0.48 |
9.189% |

0.53 |
0.43 |
0.72 |
16.409% |
0.48 |
7.876% |

0.53 |
0.43 |
0.61 |
13.902% |
0.48 |
6.673% |

0.53 |
0.43 |
0.54 |
12.307% |
0.48 |
5.907% |

0.53 |
0.43 |
0.43 |
9.800% |
0.48 |
4.704% |

WICHITA ST. |
|||||

MICHIGAN VS. KANSAS |
ELITE EIGHT |
FINAL FOUR |
EST. PROB TO REACH NC GAME |
EST. PROB TO BEAT WICHITA STATE |
CHANCE OF WINNING ON PATH |

0.53 |
0.84 |
0.72 |
32.054% |
0.64 |
20.515% |

0.53 |
0.84 |
0.61 |
27.157% |
0.64 |
17.381% |

0.53 |
0.84 |
0.54 |
24.041% |
0.64 |
15.386% |

0.53 |
0.84 |
0.43 |
19.144% |
0.64 |
12.252% |

0.53 |
0.43 |
0.72 |
16.409% |
0.64 |
10.502% |

0.53 |
0.43 |
0.61 |
13.902% |
0.64 |
8.897% |

0.53 |
0.43 |
0.54 |
12.307% |
0.64 |
7.876% |

0.53 |
0.43 |
0.43 |
9.800% |
0.64 |
6.272% |

LA SALLE |
|||||

MICHIGAN VS. KANSAS |
ELITE EIGHT |
FINAL FOUR |
EST. PROB TO REACH NC GAME |
EST. PROB TO BEAT LA SALLE |
CHANCE OF WINNING ON PATH |

0.53 |
0.84 |
0.72 |
32.054% |
0.77 |
24.682% |

0.53 |
0.84 |
0.61 |
27.157% |
0.77 |
20.911% |

0.53 |
0.84 |
0.54 |
24.041% |
0.77 |
18.511% |

0.53 |
0.84 |
0.43 |
19.144% |
0.77 |
14.741% |

0.53 |
0.43 |
0.72 |
16.409% |
0.77 |
12.635% |

0.53 |
0.43 |
0.61 |
13.902% |
0.77 |
10.704% |

0.53 |
0.43 |
0.54 |
12.307% |
0.77 |
9.476% |

0.53 |
0.43 |
0.43 |
9.800% |
0.77 |
7.546% |

ARIZONA |
|||||

MICHIGAN VS. KANSAS |
ELITE EIGHT |
FINAL FOUR |
EST. PROB TO REACH NC GAME |
EST. PROB TO BEAT ARIZONA |
CHANCE OF WINNING ON PATH |

0.53 |
0.84 |
0.72 |
32.054% |
0.60 |
19.233% |

0.53 |
0.84 |
0.61 |
27.157% |
0.60 |
16.294% |

0.53 |
0.84 |
0.54 |
24.041% |
0.60 |
14.424% |

0.53 |
0.84 |
0.43 |
19.144% |
0.60 |
11.486% |

0.53 |
0.43 |
0.72 |
16.409% |
0.60 |
9.845% |

0.53 |
0.43 |
0.61 |
13.902% |
0.60 |
8.341% |

0.53 |
0.43 |
0.54 |
12.307% |
0.60 |
7.384% |

0.53 |
0.43 |
0.43 |
9.800% |
0.60 |
5.880% |

OHIO STATE |
|||||

MICHIGAN VS. KANSAS |
ELITE EIGHT |
FINAL FOUR |
EST. PROB TO REACH NC GAME |
EST. PROB TO BEAT ARIZONA |
CHANCE OF WINNING ON PATH |

0.53 |
0.84 |
0.72 |
32.054% |
0.44 |
14.104% |

0.53 |
0.84 |
0.61 |
27.157% |
0.44 |
11.949% |

0.53 |
0.84 |
0.54 |
24.041% |
0.44 |
10.578% |

0.53 |
0.84 |
0.43 |
19.144% |
0.44 |
8.423% |

0.53 |
0.43 |
0.72 |
16.409% |
0.44 |
7.220% |

0.53 |
0.43 |
0.61 |
13.902% |
0.44 |
6.117% |

0.53 |
0.43 |
0.54 |
12.307% |
0.44 |
5.415% |

0.53 |
0.43 |
0.43 |
9.800% |
0.44 |
4.312% |

So, in a purely hypothetical world where one of these teams would be the guaranteed opponent, this is more or less what the individual paths would look like. Again, there is element of uncertainty on this side of the bracket too, if you will, but I went out on a bit of a limb and averaged these results.

MEAN |
10.571% |

MEDIAN |
9.428% |

STD. DEV. |
4.853% |

I could be wrong, but I theorize that if you took the independent paths of the other teams into consideration in this very broad hypothetical, the mean represented here would be reasonably close to the typical result you would see for the easier paths to the championship game.

Many of the more likely results would fall below this if we took everything into consideration. I like to see the mean here as more or less a ceiling in that regard. My guess is that the mean, if we actually took into account all of the estimated probabilities from the other brackets of each team surviving through their own possible routes, that the mean would fall to somewhere around the 1^{st} standard deviation below the number above (10.571%-4.853%, or 5.719%, which is fairly close to Massey’s estimate).

For example, if Louisville goes through Oregon, then Duke, then Ohio State, that 0.80 x 0.64 x 0.64 per the Massey matchups, so 32.768% for them through this path. If we combined that with Michigan’s 32.054% chance through its easiest route, it comes to 10.50% of seeing that actual matchup, but of course, through the hardest path for Michigan, we find ourselves about 2 standard deviations below the mean here at 1.41%.

If we were to follow Ohio State down an Arizona-Wichita State-Louisville route, the estimated chance of that actual result is 16.63%, meaning that in combination with our chances on the easy road, that game has a 5.33% chance of occurring. Down the hardest road, for contrast, there would be a 0.717% chance of this game occurring.

TL;DR CONCLUSION:

If you got this far, my hope is that you got the central idea here, which was to develop a construct to get a sense of what the paths to victory would be, what our chances are in a hypothetical scenario where we knew the opponent, and what they might actually be – on average – in practice. Of course, with each advancement, a layer is peeled away and odds increase, and if a team loses in the Tournament, then none of this matters. My hope was to present my own take on the “mathematical analysis” presented in several threads this week. Now, beat Kansas!

CAT PHOTO:

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