The Low Background Probability Problem
When an accurate test is used to identify an unusual event, the test will usually be wrong when it identifies an event (it will give mostly false positives). Thus, even if juries (or drug tests, or ...) are quite accurate, if guilt is unusual, most people found guilty will be innocent.
Note the shocking implications for ethnic prejudice. If the ethnic stereotype is TRUE (x group is more likely to have y negative trait), but the trait is unusual, then (a) most members of the stereotyped groups will be false accused regularly, leading to objective discrimination as well as feelings of persecution, and (b) outsiders may find the test (i.e., ethnic stereotypes) cheap, accurate and effective. The result can be massive "rational" racial/ethnic discrimination without any animus or racists.
The same will be true for health insurance. If carriers of cheaply identifiable trait x (light skin) are much more likely to get disease y (skin cancer), then a rational health insurer will decline to insure carriers of x (or charge a higher premium) even if it is also true that very few members of the group will ever get the disease. Most members of the carrier group will pay high prices even though they are not in fact likely to get the disease. Another way of saying the same thing: healthy members of the identified group (light skin) will pay for a disease that they don't have, but equally healthy members of society who do not have the cheap marker will not.
Of course, more diseases in modern societies are associated with stress, particularly the stress of low status -- so poverty and dark skin are often the relevant cheaply identifiable traits. The result is, in effect, to make healthy poor people pay for unhealthy poor people, while leaving rich folks free to pay only the cheaper costs associated with healthier rich people. Similarly, auto insurance companies often use neighborhood as a cheap test: everyone in the poor neighborhood (whether or not their car is stolen) pay higher rates than everyone in the rich neighborhood.
All insurance reduces risk by redistributing from the lucky to the unlucky, but cheap, inaccurate discriminators of this sort restrict the redistribution to only within particular arbitrary groups defined by the cheap test, not any underlying merit, behavior, or even meaningful shared characteristic. When race or poverty is used as the test, the result is to compound the problems of the unfortunate. If discrimination makes dark skinned citizens' lives harder, then profit maximizing insurance companies will rationally respond (if they are permitted to) by making their lives even harder.
The Low Background Probability Problem:Test (jury) is 90% accurate; only 10% of population is positive (negligent) |
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Tests Postive |
Tests Negative |
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Factually Postive (10%) |
.90*.10 =.09 |
.10*.10=.01 |
.09+.01=.10 That is, 10% are factually positve, of whom 90% (9% of the total population ) test positive and 10% (1% of the total population) test negative (because the test is 90% accurate) |
Factually Negative (90%) |
.10*.90=.09 |
.90*.90=.81 |
.09+.81=.90 That is, 90% are factually negative, of whom 90% (81% of the total population) test negative, and 10% (9% of the total population) test positive. |
.09+.09 = .18. That is, 18% of the total population test positive, of which 9% are true positives and 9% are false positives. That is: even though the test is 90% accurate, half of its positive results are false! |
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The Low Background Probability Problem:Test (jury) is 90% accurate; only 1% of population is positive (negligent) |
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Tests Postive |
Tests Negative |
||
Factually Postive (10%) |
.90*.01 =.009 |
.10*.01=.001 |
.009+.001=.01 That is, 1% are factually positve, of whom 90% (.9% of the total population) test positive and 10% (.1% of the total population) test negative (because the test is 90% accurate) |
Factually Negative (90%) |
.10*.99=.099 |
.90*.99=.891 |
.099+.891=.99 That is, 99% are factually negative, of whom 90% (89.1% of the total population) test negative, and 10% (9.9% of the total population) test positive. |
.009+.099 = .108 That is, 10.8% of the total population test positive, of which .9% are true positives and 9.9% are false positives. So, of those who test positive, .009/.108=8.3% are true postives. That is: even though the test is 90% accurate, 91.7% of its positive results are false! |
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