Another note from reading "How Judges Think":
Discussing whether federal district judges have incentives to shape their behavior so as to win promotion (to the Court of Appeals), Posner explains that the fact that "only 6 percent of district judges are promoted exaggerates the odds against promotion" because many district judges have no "realistic prospects of being promoted." "Suppose," he writes, "though this is just a guess, that only 20 percent of district judges had a chance of being promoted. Then each of the judges in that pool would have a one-third chance (.20 [divided by] .06 = .33) of promotion, and those might be short enough odds to induce a judge to do whatever he could to rise within the pool." (142)
This passage exemplifies Posner's pragmatic, one might at times say cynical, appraisal of human and judicial nature -- a perspective that is enlightening even if not always persuasive. But this particular passage also contains, surprisingly, a mathematical error. 0.20 divided by 0.06 does not equal 0.33; in fact it equals 3.33. (Readers whose recollection of decimals is fading can test this by adding 0.06 + 0.06 + 0.06; the result is 0.18, and the remaining 0.02 is exactly one more third of 0.06.) Since the highest chance anything can have of happening is 1 (or 100 %), clearly distict judges do not have three times as high a chance as that of becoming appellate judges. Probably that anomaly led Judge Posner to the intuitive conclusion that the answer to the calculation was 0.33, since that number is plausible.
But as it happens, that number is only accidentally plausible. The correct answer can, I believe, be approached this way. If 6 promotions are available each year, and they will all go to a group of 20 judges, what is the chance that any one of those judges will get a promotion? The answer to that question is the number of promotions divided by the number of candidates, or 6/20. (This calculation is the opposite of the one Posner did, which was to divide 20 by 6.) 6/20 = 0.3, which is quite close to the 0.33 figure that Posner mistakenly arrived at.
This is an interesting illustration of the power of intuition to undercut logical reasoning, a frequent focus of those who study human cognition.
It's also an illustration of an older proposition: even Jove nods.