The base rate fallacy

Faulty base, faulty result

There is a lot more to say about fallacies than I did in my recent blogs. It is important to avoid fallacies, for they are mistakes in reasoning and they distort the way we look at the world around us and how we get along with others and with ourselves as well! Often fallacies lead to wrong decisions or you can get unnecessarily worried about things that might happen. The book Bad arguments that I used for my lasts blogs treats hundred fallacies, but actually it’s only a selection of all the ways we can reason in the wrong manner. Sometimes one wonders how it is possible to survive, if you can make so many mistakes, but the practice is that we do, more and more successfully.

I end the present series of blogs on fallacies by discussing one that is very common: The base rate fallacy. This is the fallacy in which basic information is ignored or is confused with specific information. If you fall into this trap, you get a completely wrong image of what is happening around you or what is happening with you. You can become worried without reason, as said, for instance when you take part in a medical examination of the population and then it appears that you have a positive test result, so you may have a serious illness.

Say, a medical examination of the population is done for a certain deadly disease. 0.1% of the adult population is infected and the government thinks that it is worth to test the whole adult population of the country, for the disease can be well treated if discovered in time. Let’s assume that there are ten million adults in this country and that every adult takes part in the test. The test has a false positive rate of 5% and no false negative rate, so – besides those with a correct positive test result – 5% of the adults with a positive test result actually is not infected, while nobody who with a negative result is infected. The next step is then that everybody with a positive result is called up for further medical examinations. Now it often happens that people in this selected group think that they have a 95 chance of being really infected, for isn’t it so that the test is 95% accurate? By thinking this way these people ignore, however, that this 95% tells us only something about the quality of the test, not about the presence of the disease in the population, which is 0.1% (among adults). Therefore they may become more worried than they need to. Let me show:

- The test is applied to 10,000,000 (ten million) people and 0.1% is infected. So 10,000 people are infected. They all have positive test results.

- 5% of the tests indicate that the tested persons are infected, while actually they are healthy. So

499,500 (5% of “10,000,000 minus 10,000”) people have a positive test result, but they are not infected.

- Both the first group and the second group will have to undergo extra medical examinations in order to determine whether they are really ill or whether they aren’t. So 10,000 + 499,500 people have to undergo extra examinations, which are altogether 509,500 people.

- Only 10,000 people among these 509,500 people are really infected, so 1,96% of the selected group is really infected. Therefore, if you belong to the group selected by the first test, the chance then that you have really been infected in this example is not a high 95% but only about a tiny 2 %. Although this may be serious enough, don’t be more worried than you need to.

In his article on the base rate fallacy (see below), Manninen discusses yet another case where the base rate is ignored. In short it is this: Between 1999 and 2011, 2151 whites were killed by the police in the USA and 1130 blacks were. Therefore whites are worse off than blacks. Is it true? If you look only at the figures given here, you would think “yes”. However, according to the 2010 Census in the USA, 72,4% of the population was white and 12,6% was black. When you add this basic information to the example, the picture completely changes. Need I further explain? If you have come thus far, I assume that you are smart enough to see that in proportion to the respective populations by far more blacks were killed by the police than whites were.

There is a phrase that says “there are lies, there are damned lies and there are statistics”. However, statistics lie only because we don’t know how to use them or use them intentionally in the wrong way.

Sources,

- “Base rate fallacy”, in Wikipedia on https://en.wikipedia.org/wiki/Base_rate_fallacy

- Manninen, Thomas W., “Base rate”, in Arp, Robert; Steven Barbone; Michael Bruce (eds.), Bad arguments. 100 of the most important fallacies in Western philosophy. Oxford, etc.: Wiley Blackwell, 2019; pp. 133-136.