Bayes' theorem.

Step 1 of 5

The formula

So far we have counted people. The same sum can be written in chances. Write $D$⁠ for “ill”, $H$⁠ for “healthy” and $+$⁠ for “a positive result”. The question of someone who tested positive is $P(D \mid +)$⁠: how many of the positives are ill. The numerator is the left-hand mat, the ill positives, and the denominator is both mats together:

$P(D \mid +) = \frac{P(+ \mid D) \cdot P(D)}{P(+ \mid D) \cdot P(D) + P(+ \mid H) \cdot P(H)}$$P(D \mid +) =$$\frac{P(+ \mid D) \cdot P(D)}{P(+ \mid D) \cdot P(D) + P(+ \mid H) \cdot P(H)}$⁠

This is Bayes’ theorem. $P(D)$⁠ is the base rate, $P(+ \mid D)$⁠ and $P(+ \mid H)$⁠ are what you know about the test, and what comes out, $P(D \mid +)$⁠, is what you want to know.

Here $2\%$⁠ are ill and the test is $95\%$⁠ accurate. Work it out, then sort the positives and check.

Step 2 of 5

Turning the condition round

The denominator in the formula is simply $P(+)$⁠, all the positives, as you saw on the trees. So the theorem can be written more briefly too. Its power is that it turns the condition round: from the chance of a positive test given the disease, which a lab measures, to the chance of the disease given a positive test, which the patient needs.

Step 3 of 5

Testing again

$28\%$⁠ is still not much. What can you do? Retest the positives is on: everyone who tested positive is tested again, independently, and stays positive only if the second test agrees. Work out the chance now.

Step 4 of 5

One step at a time

The mats show what the sum said: almost all the healthy were weeded out by the second test. There is another way to see it. After the first test, the chance of being ill is no longer $2\%$⁠ but $27.9\%$⁠. That number can go into the formula as a new base rate.

Step 5 of 5

The prosecutor’s fallacy

The same mistake of turning the condition round turns up in courts too. The chance that a piece of evidence shows up for an innocent person is one thing; the judge needs the reverse, the chance the person is innocent given the evidence.