gnome said:
Probability is an attempt to glean some information from uncertainty. To complain that it doesn't give a certain answer is to ignore its purpose. There ISN'T a certain answer, but a look at the probability can help you make a wise bet, or to learn how external factors cause the tendencies to vary.
I like your first sentance a lot, but I might change it to "Probability is an attempt to provide predictability despite uncertainty."
Back to my favorite area, athletics, I like to think of an athletic competition as a probabilistic event. For example, if I took a team A and cloned it to create team B, and had them play each other, who would win (put it in a neutral arena, so that neither has home advantage
)? Since the teams are perfectly equal, you would have to see that each has an equal chance of winning. In that case, you can say that team A has a 50% chance of winning. What that means is that if they played a very large number of games, that the amount that team A would win would approach 50%. But suppose they only play 1 game? In that case, one team would win, but the other would lose. However, that does not mean that one team was actually better than the other. It is just a consequence of small sample size.
Now, suppose one team is a little better than the other. Maybe if they were to play a large number of games, team A would win 60% of the time. What would you see if you only played 1 game? Well, team A is certainly more likely to win, but we aren't surprised if team B wins. Again, a sample size of 1 is not enough to distinguish them.
Now make the difference larger. Suppose it is 90%. In that case, we would be surprised to see team B win, although it is certainly not out of the realm of possibility. In fact, if you have enough 90/10 matchups occur, you won't be surprised if the upset happens. Maybe 1 out of 10 games will be upsets, for example (of course, that is what 90% means).
Take this to the next level. Suppose a team has a 99% chance of winning. That is a very lopsided game, right? It's basically bigger than that 90/10 split above. And sure enough, you would say that a team with a 99% chance of winning is pretty clearly the better team, right? But given 100 of these types of matchups, you expect there to be an upset. That doesn't mean that the underdog is better than the team they beat, it just reflects the fact that since no one is ever 100% assured a win, given a large enough sample size, upsets will occur. In fact, I tend to say that by probability, not only CAN upsets occur, they MUST occur. No upsets occuring is far more unlikely than no upsets occuring.
This has implications in lots of places. For example, watch the NCAA basketball tournament. No #1 seed has ever lost to a #16 seed. Given the number of games that have been played (probably around 100, I think (that would be 25 years worth)), that suggests that the average 1/16 match is > 99% in favor of the #1 seed. Meanwhile, a #15 over #2 upset occurs about every other year, so it looks to be about 90%/10%. A #3 is upset basically every year, so we are talking a 75%/25% advantage there, and it keeps dropping rapidly. This is why it is so difficult to predict the outcome of this type of tournament, because the probabilities drop very quickly as you move down the seeds. By the time you are to 8 vs 9, it is basically 50/50. Now, you can make great predictions about what will unfold, and that there will be upsets in these rounds etc, but identifying where the upsets will occur is something that is very hard to do.
Another application comes in evaluating competition levels. For example, given baseball's sample size of 162 games, the final winning pct is a decent reflection of the range of quality. Thus, most baseball seasons are fairly consistent with the notion that the best team wins about 60% of the time against average competition the worst team wins about 40% of the time against average competition, and everyone else is evenly distributed between them. This will lead to standings that are indistinguishable from what is observed in real life, where the best teams win 95 - 100 (+/- a few, maybe .650 at the best) and the worst teams win 60 - 65 (+/- a few, down to maybe .350).
Meanwhile, football is very well known for it's wider distributions of records, as teams end up with records as good as 14 - 2 (.875) or even 15 - 1, and 13 - 3 is common, whereas at the other end, you see 2 - 14 and 3 - 13. However, this wide distribution is very much a consequence of a 16 game schedule. In fact, the distribution we observe for the NFL implies a scenerio where the best team wins about 65% against average competition, and 35% at the low end. The gaudy records that are obtained are actually a result of deviation in a small sample size. If baseball had a 16 game schedule and kept that 60 - 40 distribution, 12 - 4 records would be fairly common, and you would even see 13 - 3.
One consequence of this deviation in the NFL is that it is far more common that the best team (the one who has the best chance of winning against an average opponent) does not have the best record in the league, and have the third, fourth, or even worse best record. The NFL compensates for this by letting more teams in the playoffs so that the best team is more likely to get there. Of course, this means that the NFL ends up getting a lot worse teams in their playoffs, too, letting in teams that are getting there not because they are necessarily "better", but because they just happened to get wins against better teams through basically chance. If the year before they were 6 - 10 and this year they go 10 - 6, everyone thinks they had a big tournaround. However, it could just be that they are actually an 8 - 8 team that were unlucky in year 1 and lucky in year 2. The other conclusion is that despite the appearance of "parity," it is actually artificially created by having a short season and lots of playoff teams. If baseball played 16 games and had 6 playoff spots, you'd see even more teams making the playoffs, more often. Similarly, if the NFL played 162 game schedule like baseball does and had only 4 playoff teams per year, you would see the same teams dominate from year to year, even more than you do in baseball now.
This is the kind of analysis that you can do if you view the world from a probabilistic standpoint.
We can compare what our model predicts to what actually occurs.