Probability and Undefeated Seasons

BUCKfutter's picture
February 25, 2013 at 6:38p

Before I get into this - caution: nerd alert.  That said, there have been a few thread topics floating around the last few months about another undefeated season next year.  A common argument that such a season is likely seems to be "show me a game we are going to lose" or something of the like.  I agree, when taken individually, we should win every game we play next year (and, I would argue that for the last decade, save 2004 and 2011, going into each game, the majority of us would have argued that we should have won just about every game we played).  That, however, does not mean that chances are we will go undefeated.

For a 12 game schedule, even if your chance of winning each individual game is 95%, your chances of going undefeated are .95^12 = 54%.  A 95% chance of winning each game is absolutely not realistic- much too high - but the 54% overall chance GIVEN the 95% chance of winning is tellingly small.  If you take it down to a 90% chance of winning each individual game (still really high in my opinion), your chances of going undefeated drop significantly to .9^12 = 28%.  Add a B1G title game and a national championship game and you're really starting to get lower.

With that as a prologue, I am going to go through each game in our 2013 schedule, assign a percentage chance of victory (going to be very generous to OSU, as you will see), and see where we come out at the end (hint: the chances aren't great).

Game                            Chance of win

Spring Game (yee haw)   100%

Buffalo (H)                     99.9%

SDSU (H)                       95%

Cal (A)                           90%

FAMU (H)                      99.9%

Wisky (H)                       90%

NW (A)                          85%

Iowa (H)                        95%

State Penn (H)               95%

Purdue (A)                     95%

Illinois (A)                       95%

Indiana (H)                     97%

scUM (A)                        85%

Chance of undefeated reg season = 43.8%.  Again, I feel like I've been pretty generous to OSU in assigning these percentages (see: 95% chance of winning in West Lafayette AKA the Buckeye house of horrors).  We haven't even delved into the postseason yet:

Game                         Chance of win

B1G Title                        85% - rationale: again, being very, very generous.  Assuming scUM will be the best team in the legends next year - the advantage they                     gain by having seen us the week before offsets the game being at a neutral site.

NCG                              50% - assuming we play Alabama or equivalent.  Given the status of both current rosters, 50% here is more than generous to us.


43.8% * 85%* 50% = 18.6% chance of a 14-0 season, once again, using EXTREMELY generous win probabilities.

Not saying I don't want it to happen, or don't think it can happen; only by no means is it a sure thing, or for that matter, even a probable thing.  Go Bucks.


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xFactor11's picture

These numbers are pretty cool to show just how hard it is to go undefeated. I would personally put the percentages most games lower(I guess I'm not as generous). Every team gives Ohio State their best and especially after a 12-0 season, teams will want to bring us down. So theoretically, yes we should go undefeated but the chances of that happening are slim and this shows that. Great post!

mitchjacobsen01's picture

I appreciate the math and the effort you put into this.  
It puts some of these things into perspective and really helps to understand how special it is when the team goes undefeated.  I think I might just celebrate 12-0 a few more times before next season...
With that being said, there were some of us who were jumping ship and predicting an 8-4 sesason after the UCF game last year, so while we should rein in our expectations a little, on a game by game basis, the math is on our side.

xFactor11's picture

Just for fun and because I'm bored I decided to do this same type of analysis with Alabama. I believe I was also generous in my percentages given to Alabama.
Game                              Chance of Win    
Virginia Tech                    90%
@ Texas A&M*                 80%
Colorado State                 99%
Ole Miss                           95%
Georgia State                   99.9%
@ Kentucky                     99%
Arkansas                          95%
Tennessee                      95%
LSU*                              75%
@ Miss State                    95%
Chattanooga                   99.9%
@ Auburn                       95%
*Previous week is a BYE week.
Chance of regular undefeated regular season = 40.9%
SEC Championship           70%
National Championship      50%
Chance of 14-0 Season = 14.3%
This is also with very generous percentages and it is also worth noting that before their two toughest games (Texas A&M, LSU) they have a bye week. When you take that into consideration they really do not have that tough of a schedule. Although I'm sure they will still get the SEC points once the polls come out.

kevinfrenchfry's picture

call me crazy but i think ohio state could handle that schedule

xFactor11's picture

I totally agree. Who on their schedule is that tough? LSU and A&M which they have a bye week before which can drastically help. Virginia Tech is always a threat and I assume that Arkansas will be tougher. Auburn is always a rivalry game which won't be a cake walk.

NerdAlert's picture


Hovenaut's picture

Well done. Nice to see the math applied to show how difficult it is to run the table. As major college football says goodbye to the B(c)S, going undefeated will become even more difficult.

I like the schedule, I like our team and I like our coach. Meyer lit a fire back into a program that had lost its way, and he's been a man possessed this offseason.

Fully expect UFM to stoke the fires again this fall, focusing on one at a time and keeping season goals within perspective/reach.

MN Buckeye's picture

With these numbers in mind, it should make us all appreciate this past season.  What odds would you have given us in each game at this time a year ago for going 12-0?  While they are better looking toward next year for so many reasons, it helps to temper the expectations.  Hence the phrase, one game at a time!

Earle's picture

What do you suppose the probability of 14-0 was going into the 2002 season?

Have you tried Not Your Father's Root Beer?  It tastes just like the real thing, but it packs a punch (5.9%ABV).  It's a little sweet for me though.  Two is my limit.

tampa buckeye's picture

30-1 at the beginning of the year.

Earle's picture

Well, sports book odds and mathematical probabilities aren't quite the same, but using the method above and "realistic" chances on individual games, that mightve been a sucker's bet. I bet it'd be way less than 3%.

Have you tried Not Your Father's Root Beer?  It tastes just like the real thing, but it packs a punch (5.9%ABV).  It's a little sweet for me though.  Two is my limit.

BME_Buckeye's picture

A few things on this

  • I'm a nerd too so good job on this. 
  • The reason you can multiply games is because its mutually exclusive winning one game to the next.
  • I would argue that beating michigan percentage should be lower on the road. Couple that if we beat them in AA, which I think we will, a revenge factor comes up so it should drop down a little bit more. 
  • If we are being fair, the true percentage to win it all would be (.50)^14 x 100 = [some percentage that will be <1%]

Good blog! 

Look closely, because the closer you think you are, the less you will actually see.


hodge's picture

"If we are being fair, the true percentage to win it all would be (.50)^14 x 100 = [some percentage that will be <1%]"

The problem with that theory is that it assumes complete and total parity amongst competition, which is far from the case in collegiate football.  A heavy favorite--which OSU will be in many of the games it plays this year--does not have an equal chance of winning or losing the game; the favorite's overwhelming advantage in talent and depth renders an equally probable outcome impossible.  Now, if we're talking against the spread, you'd be completely correct--since point spreads are designed to handicap the outcome into an equal probability.

BME_Buckeye's picture

I did assume total parity in this case but I equally know that there about 5-6 games that are gimmies for us this year. Its hard to assign straight probabilities of winning because we don't know who will be injured, [knock on wood] its not Braxton, how much better teams will be (i.e. Northwestern), nor if curses will continue (at Purdue) but there are several variables that effects assigning preseason probabilities. 
Note to OP: I meant to say that you can use Independence, which is the reason why you can multiply the events of winning games because they are mutually exclusive. I meant to state independence. 

Look closely, because the closer you think you are, the less you will actually see.


hodge's picture

Nice analysis, I can dig it, mate.  I actually did something very similar last year, assessing the good guys' chances at perfection last year.  I was a little less generous, and wound up with a whopping 5.6% chance.  That said, I think that your B1G season predictions are a wee bit optimistic, here's my thoughts: our offense will be scoring at will, but our defense will need to pick up the slack left by the departing seniors.  With the expectation of a high-powered offense, an unproven LB corps (outside Shazier), a talented--yet green--d-line, and a good (but not yet great) secondary, these are my season predictions:

  • vs. Buffalo: 99% (garbage MAC-rificial least they're DI)
  • vs. San Diego State: 95% (9-4 is still 9-4 last year, even if they beat terrible teams)
  • @ California: 90% (hey, jet-lag is scary stuff)
  • vs. Florida A&M: 99% (walking into a buzzsaw is never fun)
  • vs. Wisconsin: 80% (their style of play will test an untested defense; thankfully we get them at home, else they might be an upset special)
  • @ Northwestern: 69% (Fitzgerald's got these guys on the up-and-up; and with a bye week heading into this game, their B1G opener, they're going to be jacked)
  • vs. Iowa: 89% (coming off a bye, our biggest challenge will be avoiding complacency)
  • vs. Penn State: 82% (O'Brien's system might mold Hackenburg into a threat by this point of the season; this is a talented, albeit young, team)
  • @ Purdue: 87% (I'm betting that Urban doesn't let the team forget that the Boilermakers gave them their closest shave last year)
  • @ Illinois: 79% (I'm expecting the Illini to make a turnaround in Beckman's second year, this could be a game)
  • vs. Indiana: 88% (defense should come out hungry to redeem last year's infamous performance)
  • @ Michigan: 49% (I really wouldn't be surprised to see Michigan favored here, if by only a smidgen; the Buckeyes will face their only real "marquee" matchup of the regular season--which will point to us as being largely "untested" in a big game--as Devin Gardner will in the midst of a good-to-great season, and the Wolverines will probably enter with only one or two losses)
  • Big Ten Championship: 58% (either Nebraska or Michigan, this game will be tough, but I expect OSU to have a crowd advantage in Indiana)
  • BCS National Championship: 45% (Vegas likes 'Bama and then Oregon, and either would create one hell of a matchup, but 'Bama is an all-around better team than the Ducks, especially considering that Oregon has struggled against teams proficient in stopping the run; I'll split the difference between the favorable and unfavorable matchup)

With those (way too damn early) odds out there, I'd give our Buckeyes almost an 11% chance at making through the regular season unscathed, and a 2.8% chance of making a run at perfection in 2013.
Again, it's worth mentioning that Vegas gives us 6:1 odds as national champs.  That means that they see us as having a 16.66% chance of going undefeated (since odds are that we wouldn't be the first one-loss team in, considering our schedule).  I see the discrepancy as Vegas only counting six games with OSU really standing a good chance at losing, which I see as Northwestern, Michigan, B1G Championship, and the BCS National title, along with two other unforeseen "upset" games (like Purdue, Cal, and Indiana last year, heavy underdogs who gave the favorite a run for their money), which I'm thinking will be Illinois and Wisconsin.  I tend to think the latter model is a bit more accurate, since the vast majority of these contests will not be close, the Vegas model basically predicts that OSU's schedule and aggregate talent will yield six "contested" games that the Buckeyes will stand a decent chance of losing; our "percent chance of win" predictions are almost forecasting which games have the greatest probability of a contested result.

kevinfrenchfry's picture

SOOOOO, if everyone is doing probabilities, what was the probability of going 12-0 this past season, because i would've put very few games above 65 percent from 2012

Run_Fido_Run's picture

Nice blog post, Buckfutter.
I came up with 18.3 percent chance of the Buckeyes going 12-0 during the regular season; 13.7 percent chance of exiting the BT CG 13-0; and 6.1 percent chance of finishing 14-0.
Usually, when I run these types of numbers I get even worse chances, but I think Hodge is on the right track:

  • If we assume that 2013 Ohio State will be an elite team, then the chances that the Buckeyes will beat Buffalo, FLA A&M, or SD State approach 100.0 percent. Likewise, the chances that Ohio State will beat decent teams like Iowa, PSU, Indiana, ILL (which, theoretically, could win on "any given Saturday") maybe reach 94 - 97 percent. Thus, it comes down to a handful of road games and/or games against pretty good opposition - at Cal, Wisc, at NW, at Purdue, Michigan.
  • On the other hand, if 2013 Ohio State does not turn out to be elite, several additional games will be "in play," and the chances of going undefeated would drop precipitously. But were not really asking what are the chances that a pretty good team goes undefeated, are we?

What has me a bit perplexed, though: What if prior to Nebraska's 1995 season we'd guessed the chances they'd go undefeated during the regular season? They played some good teams, including Kansas St, Colorado, Oklahoma (which was in a down period, but still a rival). If we'd run the numbers, maybe we'd have guessed that NU's chances of going 11-0 were like 35 percent, which is very high. Yet, in retrospect, we discovered that their "true" odds were much better than that. Nebraska did not play a close game all year and blew the doors off several top 10 teams. The '95 Cornhuskers probably go undefeated 7 out of 10 times against that schedule, but how do we explain that using this methodology?
I'm by no means comparing the 2013 Buckeyes to the 1995 Cornhuskers. But '95 Nebraska example does make me wonder if running these types of numbers leaves out something from the analysis.       

BUCKfutter's picture

I think in the case of a team that dominant, their average win probability is above 95%, putting their p(undefeated) over 50%.  As you said, tough to discern if a team is going to be that dominant before we watch them for a few weeks at least.

the kids are playing their tail off, and the coaches are screwing it up! - JLS

causeicouldntgo43's picture

" Very sound analysis Trebeck, your mother hypothesized the same theory to me last night".....: said with my best Sean Connery accent (from the SNL Jeopardy skit).

JLBNYC's picture

When i signed up for 11W, I was told there would be no math!

hodge's picture

Okay, so I did some thought last night, and I think I've got a better formula.  The question I've been asking is, "Why are Vegas's predictions so much more optimistic than our analysis?"   know I think that we've got a better than 2% chance of making it through the season completely unscathed, but why does our math say otherwise?
he problem with a mere "expected win probability" is that it doesn't evaluate the probability that such a prediction will actually be accurate.  For example, when OSU plays Northwestern, I'm thinking that we've got about a 70% chance to win that game, because the 'Nerds are going to pose a tough threat to us.  But that 70% evaluates the probability of OSU winning a game in which both teams play consistent to the way that they've played all season.  What if Northwestern or Ohio State lays an egg?  How do we account for the probability of this kind of thing happening?  
This led me to the idea of using a "talent differential" to give a better picture of how a lesser team is to honestly have a chance at beating us.  This measure indicates the disparity of talent (a subjective measure), and applies that ratio to the "loss probability" (merely the inverse of the "win probability"); thereby giving us an idea of the probability that a lesser team could play up to its potential against a more talented team like Ohio State.  Naturally, this measure creates large disparities between great teams and lesser teams, and only slightly adjusts predictions betwixt more closely-matched teams.  My method is below.

AWP [adjusted win probability] = 1 - (TD [talent disparity] x LP [Loss Probability])
LP = 1 - WP [win probability]
1 - (TD x  LP) = 1 - (TD x (1 - WP))
TD is composed of dividing your opponent's talent measure (an arbitrary constant of 1-100, with 100 being best), To, by your team's, T1, and then subtracting that number by .  TD = (To / T1)
Therefore, our adjusted win probability formula is as follows:
AWP = 1 - ((To / T1) x (1- WP))

For example, when Ohio State plays Michigan in Ann Arbor, I'd wager that we might be a slight underdog; therefore I'll say that we have a 49% chance of victory.  But, to determine the talent disparity I must rate each team subjectively, so let's say that OSU is a 90 and Michigan is an 84.  This shows that Ohio State is more talented, but such a slight edge will only result in a small adjustment of our prediction.  Let's crunch some numbers!

To = 84 [Michigan's talent rating]
T1 = 90 [Ohio State's talent rating]
WP = .49 [Ohio State's win probability]
1 - ((84 / 90) x (1 - .49)) = .524

According to this measure, Ohio State's slight talent advantage helps to mitigate their 49% chance at victory, leading to an Adjusted Win Probability of .524 or 52.4%.

hodge's picture

Here's the full season's AWP, I'm going to grade Ohio State's talent (T1) at 90 for all my formulas:

  • vs. Buffalo (10 [To]), 99% [WP]: 1 - ((10/90) x (1 - .99)) = .999 or 99.9% AWP
  • vs. San Diego State (52), 95%: 1 - ((52/90) x (1 - .95)) = .971 or 97.1% AWP
  • @ California (45), 90%: 1 - ((45/90) x (1 - .90)) = .95 or 95% AWP
  • vs. Florida A&M (5), 99%: 1 - ((5/90) x (1 - .99)) = .9995 or 99.95% AWP
  • vs. Wisconsin (72), 80%: 1 - ((72/90) x (1 - .80)) = .84 or 84% AWP
  • @ Northwestern (78), 69%: 1 - ((78/90) x (1 - .69)) = .732 or 73.2% AWP
  • vs. Iowa (49), 89%: 1 - ((40/90) x (1 - .89)) = .951 or 95.1% AWP
  • vs. Penn State (69), 82%: 1 - ((69/90) x (1 - .82)) = .862 or 86.2% AWP
  • @ Purdue (59), 87%: 1 - ((59/90)) x (1 - .87)) =.915 or 91.5% AWP
  • @ Illinois (65), 79%: 1 - ((65/90)) x (1 - .79)) = .849 or 84.9% AWP
  • vs. Indiana (55), 88%: 1 - ((55/90)) x (1 - .88)) = .927 or 92.7% AWP
  • @ Michigan (84), 49%: 1 - ((84/90)) x (1 - .49)) =.524 or 52.4% AWP
  • Big Ten Championship (84), 58%: 1 - ((84/90)) x (1 - .58)) = .608 or 60.8% AWP
  • BCS National Championship (95), 45%: 1 - ((95/90)) x (1 - .45)) = .42 or 42% AWP

Using these AWP ratings, we have an aggregate chance of 17.5% to make it through the regular season unscathed (compared to 11%, unadjusted), and an aggregate chance of 4.4% to win it all (compared to an unadjusted chance of 2.8%).  Still, not quite Vegas's 16.666%, but Vegas's numbers also account for the chance that no one goes undefeated and OSU backs into the title game; whereas mine are based solely on a perfect season.  Sounds about right, as even the most elite teams have difficulty maintaining perfection.

Run_Fido_Run's picture

Hodge, you offer some interesting ideas on how we might close the apparent (predictive) gap between: A). Pre-season expected win probabilities; and B). The actual chances an elite team will go undefeated in X-#games against Y-schedule (keeping in mind the other factor, which is that Vegas is looking to make a profit and therefore is offering Ohio State at a "bad price," so to speak).
If we know/assume in advance that we're dealing with an elite team, we probably have to make some upward adjustments, as you attempt to do. After all, what's the point in projecting the chances that a sort-of-good team goes undefeated?
Whereas, what if we begin from the assumption that 2013 Ohio State is an undefeated-caliber team (UCT) against a generic schedule? Perhaps, hypothetically, a UCT will go 12-0 against an "average" schedule about 25 percent of the time. Well, maybe Ohio State's schedule is weak enough that, as a UCT team, they should be projected to go undefeated 35 percent of the time?
Or, we could work backwards. Let's assume that Ohio State goes 12-0 in 2013. That doesn't mean, however, that it was fate that they went undefeated. They'll probably have won some close games and maybe even one to two that could have gone the other way if not for a worse bounce, less favorable officiating, etc. In retrospect, knowing that they were an elite team, how would we then go back and assign probabilities to each game?
I'm guessing that for some of the games against the likes of at Purdue or at Illinois, if/when we'd assigned the chances before the season, we'd have guessed something like 82 percent or maybe even 78 percent, etc. In retrospect, though, we might decide that 2013 Ohio State was going to win these games 92+ percent of the time.   
For example, we might look back at 2006 Ohio State. Never before the season would we have assigned such favorable chances as follows, but maybe they make sense in retrospect:
Northern Illinois: 0.995
at #2 Texas: 0.760
Cincinnati: 0.990
Penn State: 0.920
at #13 Iowa: 0.910
Bowling Green: 0.995
at Michigan State: 0.960
Indiana: 0.990
Minnesotada: 0.970
at Illinois: 0.900
at Northwestern: 0.970
#2 Michigan: 0.680
Chances of 12-0 = 0.341

hodge's picture

I agree on the general fallacy of attempting this kind of analysis without any body of work, but it's all in the name of good fun.  I do believe that, if we were to put a great deal of effort into determining these variables for our opponent each week, we'd have a pretty good snapshot of not just the team's abilities, but also for level that they were playing at when we faced them.  Then, at the end of the season an aggregate tally could be made, and perhaps compared to in-retrospect analysis, like the one you made for the '06 team.
The only problem with doing such things in retrospect, is that you lose track of the overall momentum that the team builds/loses during the season's run.  For example, OSU started last year moderately hot, then cooled with close (or closer than expected) wins over  UCF, UAB and Cal.  Our momentum then turned after a close victory in East Lansing, and a shellacking of Nebraska.  But then close shaves against Indiana and Purdue checked our momentum again, en route to a big win over Penn State, of which we rose that momentum through to a big win over Illinois, and gutsy victories over both Wisconsin and Michigan.  Therefore, our chances of beating Michigan after defeating Wisconsin would have been much higher than, say, had we played them right after the Cal or Indiana game.

Run_Fido_Run's picture

I agree, completely, that these predictions are all in the name of good fun. This was just a question that was puzzling me for some time. I would caution homer-istic Buckeye fans, like BuckFutter does in this blog post, "Now, remember, the chances that even a great team will go undefeated might only be . . ."
Yet, the non-mathematical side of my brain (which is probably 99.97 percent of it) was skeptical. "How come teams like '95 Nebraska, '02 Miami, etc. barely had to break a sweat during their regular seasons? And if it was that easy for them, why are my back-of-the-envelope numbers suggesting that Ohio State has only a 9 percent chance of going undefeated against a weak ass 2013 schedule . . .?"
Last year, for example, my numbers projected that Ohio State would win about 8.8 games, or thereabouts. But I didn't believe the numbers and instead predicted 10-2. Even that was too pessimistic, though, as it turned out. Sometimes our instincts are better than our quantitative analysis (although others times are instincts can be woefully, tragically wrong!).

hodge's picture

I think that a lot of this aforementioned disparity is due to the "win percentage" estimation.  Perhaps it's too subjective, and we need a more quantifiable metric, as opposed to just rough estimation?  Actually, the more I think about it, Accuscore's simulations are expressed in probability, aren't they?  Granted, their formula probably takes talent disparity (amongst many other variables) into account.

BUCKfutter's picture

This is fantastic.  Nothing better than nerdy glory on a boring Tuesday

the kids are playing their tail off, and the coaches are screwing it up! - JLS

hodge's picture

The sad thing is that I took maybe two math classes in college.  I was a Strategic Communications major with a minor in Political Science.

BrewstersMillions's picture

That's not sad at all. Good for you that this stuff comes easy despite 'formal' training. I'm marrying a math teacher and she took some really advanced stuff in college at Iowa-and like actual advanced stuff, not just the normal one plus one equals two stuff most Hawkeyes consider "advanced" and my head spins.
I was an English major at OSU. The relationship between numbers and letters is very clear to me. Why the two parties have to mingle with one another is beyond me.

Run_Fido_Run's picture

I studied history and avoided every math and "hard" science class like it was the plague. I didn't start appreciating math until I got into sports handicapping and betting on the ponies. By then, it was too late for a mathematically-stunted bum like me.

jedkat's picture

but can you guys help me do my taxes?

"Can we please stop the message board fighting? I really can't stand the message board fighting..."

"No. You're an idiot, and your posts are terrible."

harleymanjax's picture

My prediction for 2013
Alot depends on TTUN, if they make the B1G championship game, I think it will be hard to beat them 2 weeks in a row. So we finish with 1 loss and do not make the NCG.
If TTUN does not make it to the B1G CG, I say we finish undefeated and lose to Bama in the NCG, followed by a National Championship in 2014

"Because I couldn't go for 3"

Phillips.449's picture

Probability that the buckeyes went 12-0 in 2012======>100%!!!
To your point let's celebrate some more! (nice friggin' thread by the way!  It certainly made me think!)

MiloD20's picture

There is some bad math here.  I don't know if future games are listed in Vegas yet but if I had that info we could get a real accurate win percentage for each game.  We cant just arbitrarily make up numbers and expect our percentages to be even close.  The spread is really accurate at depicting the probability of winning each game.  
Once we have that info it is much harder to predict the chance of playing in the title game and winning it all.  There are two reasons why vegas odds are much more optimistic that anything that has been listed on this site.  First, it would be dumb for Vegas to list the actual odds because they know stupid bettors will think osu at 7-1 is a steal when in reality it is much closer to 20-1 to win it all.  Vegas just prints money on these future bets.  They are probably the most -EV bets in Vegas.  Second you will need a database to run a monte Carlo simulation to determine what percentage of the time OSU gets in the title game if they have 1 loss, or even possibly, 2 losses.  Any percentage we calculate on here isn't quite close enough because there is a decent chance OSU gets in with one loss and that is pretty hard to account for because that depends on Bama, Oregon, OU, Florida, etc etc.   

causeicouldntgo43's picture

Sure is nice to be talking about the probabilities of CONTINUING  to be undefeated next year! Nice work guys. Just thinking about this deserves/requires another snort of Elmer T Lee single barrel......................Ahh, that's the ticket................