Mathematical Way of Thinking Rationally

We often think that strong evidence means something must be true. But that’s not always the case. For example, if you have a test for a disease that is 90% accurate, it little tells you that you really have the disease on its own.

Bayes’ theorem is a mathematical formula that helps us update our beliefs based on new evidence. Instead of taking a single clue at face value, it shows how to combine what we already know (our prior belief) with new information to get a more realistic probability of an outcome. You can read the formal definition here on Wikipedia

New evidence updates prior beliefs, it doesn’t generate them out of nowhere. To decide whether you actually have the disease, you need additional information like how common the disease is in the population. That’s your prior belief, and only by combining it with the new evidence can you get a realistic answer.

P(Evidence) = P(Evidence | Hypothesis) × P(Hypothesis) + P(Evidence | ¬Hypothesis) × P(¬Hypothesis)

We have:

• Disease prevalence: 1% → P(Disease) = 0.01
• Test accuracy if diseased: 90% → P(Pos | Disease) = 0.9
• False positive rate: 10% → P(Pos | No Disease) = 0.1

Update your belief (posterior)

P(Disease | Pos) = ( P(Pos | Disease) × P(Disease) ) / P(Pos)
P(Disease | Pos) = (0.9 × 0.01) / 0.108 ≈ 0.083

Even with a positive test, the chance of having the disease is only 8.3%.