Reversing a conditional probability

The most examined idea in the standard

• $P(A \mid B)$ and $P(B \mid A)$ are different numbers answering different questions, and treating them as the same is the most common serious error in probability — in exams and in public life.

$$P(A \mid B) \neq P(B \mid A)$$

• Two statements that sound alike but are not:
  • "Of people with the disease, 99% test positive." → $P(\text{positive} \mid \text{disease})$
  • "Of people who test positive, 99% have the disease." → $P(\text{disease} \mid \text{positive})$
• The first can be true while the second is wildly false. Which one holds depends on how common the disease is.

Why the base rate decides everything

• The reason the two differ is that they have different denominators, and those groups can be wildly different in size.
• If a disease affects 1 person in 1 000, then out of 1 000 people:
  • the "has the disease" group has 1 member
  • the "tests positive" group may have 50, because a small error rate applied to 999 healthy people produces many false positives.
• A small percentage of a huge group beats a large percentage of a tiny one. This is the whole idea, and it is worth carrying as a sentence.

The method: natural frequencies

• The reliable way to reverse a conditional probability is not a formula. It is to imagine a large population and count.

• The four steps:
  • 1. Choose a convenient population — 10 000 or 100 000 works well.
  • 2. Split by the condition using the base rate.
  • 3. Apply the conditional probabilities within each branch.
  • 4. Build the group you are conditioning on and take the fraction.
• This turns a confusing problem into arithmetic you can check, and it makes the answer explicable — which matters, because these questions almost always ask for an interpretation.

The vocabulary of testing

• Sensitivity and specificity are properties of the test. The positive predictive value is not — it depends on how common the condition is in the group being tested.
• This is why the same test performs very differently as a screening tool (whole population, low base rate, many false positives) and as a diagnostic tool (people already showing symptoms, higher base rate, far fewer false positives).

The formula, if you want it

• The reversal can be written as:

$$P(A \mid B) = \frac{P(B \mid A) \times P(A)}{P(B)}$$

• where $P(B)$ comes from the law of total probability. You are not required to use this form, and natural frequencies are both safer and more explicable. Use the formula only as a check.

Where this appears outside exams

• Medical screening — the classic case above.
• Drug and alcohol testing in workplaces, where a rare true rate makes most positives false.
• Security and fraud screening, where flagging rare events produces overwhelming false alarms.
• Courtroom reasoning — confusing "the probability of this evidence if the person is innocent" with "the probability the person is innocent given this evidence".
• Everyday reasoning — "most burglars wear dark clothing" does not make most people in dark clothing burglars.