[Book] False Positives and Positive Fallacies

Conditional Probability

We learned earlier that more details can lead to more restriction and reduce the corresponding probability. More specifically, the probability of event A is always greater than the probability of events A and B.

But details do not always reduce the probability. It can increase the probability by shrinking the sample space. We can be more sure that a specific outcome will occur if we can cross off some of the sample space. This is known as “Bayes’ Theorem”.


Bayes’ Theorem

If you recall, Bernoulli considered the question

“Given a specified number of white and black balls in an urn, what is the probability of drawing a black ball?”

There is a minister and mathematician, who was interested in the following problem

“Given that one or more balls has been drawn, what can be said about the number of white and black balls in the urn?”

His name is Thomas Bay. He coined two terms.

• Prior probability: The basic probability or initial estimation of an event
• Posterior probability: The probability of the same event, when given more information (e.g., some of the balls are removed).

False Positive

Consider an HIV testing scenario. With 10,000 test subjects, suppose the result is shown below:

Suppose your blood test is HIV positive. What is the chance that you are not infected? It’s not uncommon to think that the chance is pretty slim (i.e., 10 in 10,000 or 1%). And, you might think that you will not live long enough to see your grandchild.

This is another fallacy. Look at what you are considering. You are comparing false positive test to all test subjects regardless of their test results. What you should really looking for is as follows

“Among those whose blood test is positive, how many people are not infected?”

Then, you will see that the error is 10 (false positive) in 10+1 (blood test is positive). The error is around 91%! On the other hand, there is no chance (0), that you will be infected if you blood test is negative. So you might consider taking the test again, if you blood test might turn out positive. But you do not need to take another test if your blood test is negative. The test result per se is necessary but not sufficient. We need to factor in the relevant sample space.

Fallacy: Prosecutor’s Fallacy

Prosecutor’s fallacy is statistical misconceptions when making legal arguments. Common prosecutor’s fallacy includes, for example, drug test, and DNA test.

Source: The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow, Chapter 6 [Read the Book Review] [Read the Previous Part] [Read the Next Part].

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Book or Audiobooks?

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