Tuesday, August 21, 2012

Expectations and Probabilities

Peyton Manning is a quarterback, primarily for the Indianapolis Colts (a team I strongly dislike), now with the Denver Broncos. He is one of the very best ever at what he does (coming back from behind late in games). Because of that he has hurt my team many, many times in his playing career, and likewise any time he fails to do so is cause for an unusual amount of celebration. He was injured last season, and while he will be back for this coming season, we still do not know if he will ever be the same again.

Mariano Rivera is a closer for the New York Yankees (a team I strongly dislike). He is, almost without question, the very best ever at what he does (holding small leads in the late innings). Because of that, he has hurt my team many, many times in his playing career, and any time he fails to do so is cause for an unusual amount of celebration. He is injured this season, and while his will likely be back next season, we still do not know if he will ever be the same again.



On the whole, these two are extraordinarily similar from my perspective, given the different sports they play. But there is one key diference, and the injuries each of them have suffered recently bring that diference into sharp relief. While I once really hated Manning (his previous injury and some inner soul searching have left me unclear on what I really want to happen to him, although I am quite sure I will not root for him), I have never felt anything more than impersonal frustration about Mariano Rivera.

One possibility is that Rivera is simply more classy and human seeming than Manning. There is something to this: Manning seems to have been in more television adds than anyone in his generation, and he is known making the "Manning face" and occasionally blaming his offensive line when frustrated. Rivera, meanwhile, is not in very many national commercials at all (Part of this is no doubt that I like in Red Sox territory, but I have seen far more Derek Jeter endorsements), and his extraordinarily classy behavior when he was cheered by Red Sox fans the first time they saw him in 2005 (thanking him, of course, for a number of critical postseason losses which made the Red Sox incredible comeback and subsequent 2004 Championship possible) is hard to hate on. But I think another key reason is something more centered around our perceptions of probability.

Dungeons and Dragons is a role playing game. It has a bunch of rules, none of which I have any interest in relaying here (not even the ones that I know). I bring it up because much of what is done is centered around rolling a twenty-sided die, and it makes a fine example. Rolling a twenty (a "critical") is very, very good, while rolling a one (a "fumble") is similarly bad. If you roll a twenty or a one on any given roll, you consider yourself very lucky or unlucky as appropriate, and for that given roll you were. However, if you make forty or so rolls over the space of a couple of games and do not roll any ones (or twenties, assuming normal distribution of the other extreme), you have actually been quite lucky, as we would have expected you to roll roughly two ones over that time. However, you probably would not notice this, as an extreme result like a one or a twenty is simply not expected to happen on any particular one of those forty or so rolls.

This, in my opinion, is the real key to my personal opinion on Manning and Rivera. Rivera is a closer: he is usually brought in to protect an existing lead for one inning, perhaps more in some cases. An average closer facing a relatively difficult save situation (up by one run at the start of the inning) will still hold the lead and earn a save 78% of the time. Rivera is much better than an average closer, and over time the diference can be very, very big, but for any given instance the diference between that 78% and Rivera's success rate (~85%? I do not actually know beyond his save percentage in all situations, which is 89%) is not really noticeable.

By contrast, an average NFL team in a typical 2-minute drill situation (down by three, fourth quarter, 2:00 remaining with the ball on their own twenty) has a roughly 17% chance of winning. 17%. Plenty low enough for that to be far from the expectation, but Manning (or Brady, who I do not hate because he is awesome for my team) seems to make those with mechanical precision, turning a highly improbable situation into the expectation, which is inherently more frustrating and more noticeable than making the already improbable less likely.

This actually connects back to the Red Sox struggles. An interesting topic I found perusing this SOSH thread was that the reason the Red Sox have such a poor record, despite what looked like contender level talent coming into the season (note that injuries and terrible ineffectiveness by some of that talent were also significant contributors), is caused in large part by an inability to hit against opposing closers. They have had their share of blown games, but for whatever reason (be that small sample size luck or players not suited to hit top notch reliefers) they have not had the matching share of comebacks. Of course, we do not notice that (or I certainly did not) because we don't really expect a huge and improbable comeback in any one game, and we follow the games individually, not as one big data set. I found this idea fascinating, and it inspired the rest of this post.

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