Officials in Oz are concerned about a disease that affects the population. The disease is serious but can be treated if caught early. It would be natural to test everyone right away, but tests are not foolproof: the chance that a sick person will test negative is 2%, and the chance that someone healthy will test positive is 3%.
False positives are especially problematic: in addition to causing the person the anguish of thinking that their life is at risk, they also point to expensive, uncomfortable and, in this case, unnecessary treatment. But the chances of error seem pretty small, isn’t it worth taking the risk anyway?
A central question is the following: when the result is positive, what is the chance that the person is healthy and therefore the treatment is not justified? The answer is given by the theory of inverse probability, created in the 18th century by the work of the British Thomas Bayes and the Frenchman Pierre-Simon de Laplace (the expression “inverse probability” was first used in 1837 by Augustus de Morgan, another British ).
An estimated 1% of the population of Oz is infected: this is the probability that a person chosen at random is sick. We write this in a simplified form: P(sick)=1%, so P(healthy)=99%. We want to know what the probability P(healthy if positive) is that the person is fine, knowing that they tested positive. Bayes’ theorem explains how to calculate, and the result may surprise you.
The first step is to calculate P(healthy) times P(positive if healthy). Since P(healthy) is 99% and the chance of false positives is 3%, this gives 99% times 3%, which is 2.97%. The second step is to do the same calculation for sick people, that is, P(sick) times P(positive if sick). Since P(sick) is 1% and the chance of false negatives is 2%, this calculation gives 1% times 98%, or 0.98%.
The last step is to divide the first of these two numbers by the sum of both, that is, P(healthy if positive) is equal to 2.97% divided by 2.97+0.98%. The result of this division is 75.2%. So, in this case, the vast majority of positive results —more than 3/4 of them— are false positives!
This is not to say that the government should necessarily give up on the idea of testing the population. It just goes to show that the test results need to be interpreted with caution.
I am Janice Wiggins, and I am an author at News Bulletin 247, and I mostly cover economy news. I have a lot of experience in this field, and I know how to get the information that people need. I am a very reliable source, and I always make sure that my readers can trust me.
