Wednesday, September 04, 2013

The Inspection Paradox

Amir Aczel has a very nice post on the inspection paradox, which deals with phenomena of the sort we usually file under watched-pot-never-boils -- things like the fact that if you have to wait at a bus stop you usually have to wait a long time. If you waited the entire time, you would find that the average wait time for the entire day (for instance) is a certain amount of time, but if you're coming in during the day you'll probably have a wait time longer than that overall average, because you're more likely to come to the bus stop during a long wait period than a short one. I very much like the immigration example: immigration can, in and of itself, increase a nation's average life expectancy. The reason is that people can't age backwards. If you move to a country at the age of 30, you have increased by one the number of people in the country who will live to be older than 30. If you move to a country at the age of 50, then there is one more person in the country who cannot possibly die before 50, another person who is at least 50 and therefore did not die before that age. So Israel, with a very large number of immigrants, gets a boost in its life expectancy because it has lots and lots of people constantly coming in who cannot possibly die at a very early age, because they are already older.


  1. David_LloydJones2:46 PM

    Well put -- but I don't see any paradox here.


  2. branemrys8:49 PM

    Well, it's the longstanding name of the phenomenon, whether there's an actual paradox or not. But the paradox arises from the fact that on average, you get numbers larger than average, under certain conditions. For instance, if people go to the bus stop, it can be guaranteed that their wait-times will usually be greater than average, as long as some relatively minor mathematical conditions obtain, or, put even more loosely: the average person will have longer wait times than average.

  3. David_LloydJones1:03 AM

    All of the averages are different from the average of all.
    Absence of evidence is not evidence of absence.

    Apothegms, not paradoxes.

    Here's your apothegm for "oxymoron," which many people use incorrectly as though it meant contradiction:

    Oxymoron is to contradiction as synthesis is to antithesis.



  4. branemrys8:47 AM

    It's not that all averages are different from the average of all but that all the averages can be larger than the average of all that makes the paradox.

    A paradox, it should be noted, is not (necessarily) a contradiction, either.

  5. David_LloydJones9:47 AM


    par·a·dox (pr-dks)


    1. A seemingly contradictory statement that may nonetheless be true: the paradox that standing is more tiring than walking.

    2. One exhibiting inexplicable or contradictory aspects: "The silence of midnight, to speak truly, though apparently a paradox, rung in my ears" (Mary Shelley).

    3. An assertion that is essentially self-contradictory, though based on a valid deduction from acceptable premises.

    4. A statement contrary to received opinion.

    [Latin paradoxum, from Greek paradoxon, from neuter sing. of paradoxos, conflicting with expectation : para-,beyond; see para-1 + doxa, opinion (from dokein, to think; see dek- in Indo-European roots).]

    And as you can see simply by doing the arithmetic, there is no paradox involved in the difference, of individual averages from their smaller group average.

    The phenomenon you have pointed out is called a fact. An interesting fact, but one without a smidgen of paradox to it.




  6. branemrys4:23 PM

    The freedictionary is hardly an especially authoritative source, but your appeal to it is somewhat baffling since it quite clearly shows by definitions 1 and 4 that (1) facts can be paradoxical (they just have to be seemingly contradictory or contrary to received opinion); (2) paradox is not synomous with contradiction, nor are all paradoxes contradictions; and (3) the fact that something is well-defined mathematics cannot be a good reason for denying that it is a paradox because mathematics does not deal with two of the things that can make soemthing paradoxical: the way things appear to people and received opinion about them. I have no idea why you think an appeal to a source that is inconsistent with your argument somehow makes your case.
    Further, I am not sure why you think this is important to argue. As I noted in my first comment, I am not the one who gave it the name 'inspection paradox'; it's the standard name for the phenomenon, and is used even by mathematicians who study renewal theory and are quite a bit more familiar with the underlying mathematics (which takes more than arithmetic to prove, incidentally, because the phenomenon cannot even be formulated in weaker mathematics than advanced probability theory capable of covering Poisson processes -- this is why there are professional mathematics who study the inspection paradox ) than you or I. If you don't like the usage, I'm not the one you should be arguing with.


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