It’s well known that our intuitive approaches to probabilistic reasoning lead to fairly bizarre beliefs and behaviour in some circumstances. It can also lead to fairly odd attitudes and emotions in the right circumstances. Consider, for example, how it would feel being a fan of the various teams in the American League playoff race.
Baseball Prospectus has introduced a new model calculating the probability of each team reaching the playoffs given their current standings, their performance to date, and their upcoming schedule. I don’t know how good the model is, but let’s assume for now it’s accurate. If so, here’s the probability of each team making the playoffs as of the morning of August 30.
Red Sox 73.0%
White Sox 46.2%
From that report you’d think Mariners fans should be at least as happy about their position as White Sox fans, maybe more so. But I suspect that’s not the case right now, because of the odd way the playoff teams are chosen.
There’s four playoff teams – the winners of the eastern, central and western divisions, and a wildcard for the best second place team. In effect the best three of the Yankees, As, Red Sox and Mariners will go through, two as eastern and western winners and the third as wildcard. And the best of the White Sox, Twins and Royals will go through as winner of the central.
Right now in the four team race for three spots, the Mariners are looking by far the weakest of the four. It’s pretty close, but it’s much easier to see the Mariners missing than any of the other three. It would be hard to feel particularly happy about your position if you’re a Mariners fan, because one of the four has to miss and you’re the most likely one to do so, by far.
The three team race for the central crown is tighter, but the White Sox have the upper hand right now. It would be easy to feel confident if you’re a White Sox fan. One of the three teams has to go through, and it’s most likely going to be you.
It should be clear what went wrong in the reasoning in the last sentence of each of the last two paragraphs. When there are many possible outcomes, you shouldn’t pay as much attention to which of them is most likely as much as to how probable each of them is. If Mariners fans did that they would think “I don’t know who we’re going to finish ahead of, but I’m still confident enough we’ll finish ahead of someone.” And if White Sox fans did that they might think “I don’t know who’s going to beat us, but it’s still a good chance that someone will beat us.” (That might be too depressing – it’s better if you’re a fan to focus on the positive sometimes. It’s possible to be too rational in sports sometimes. So let’s focus on why Mariners fans might be too depressed.)
The fallacy here, assigning too much weight to the most likely outcome, is I think reasonably common. But even once you’ve seen it it’s hard to overcome it. If I were a Mariners fan, I’m not sure I’d find the reassuring speech in the last paragraph that reassuring, even if I could (at some level) recognise the soundness of the reasoning.