A disease affects 1% of the population. A test for it is 99% accurate (99% true-positive rate, 99% true-negative rate). You test positive. What is the probability you actually have the disease?
AAbout 99%, since the test is 99% accurate
BAbout 50%, because the low base rate matters as much as test accuracy
CAbout 1%, since the disease is rare
DEssentially 0%, since false positives dominate
Bayes' theorem: P(disease|positive) = (0.99 × 0.01) / (0.99 × 0.01 + 0.01 × 0.99) = 0.0099 / 0.0198 ≈ 50%. The intuition: the test produces roughly equal numbers of true positives (1% of the population × 99% detection) and false positives (99% of the population × 1% error rate). This result shocks most people because they confuse test accuracy with post-test probability.
Question 2 True / False
The probability of observing evidence E given hypothesis H is typically equal to the probability of H given evidence E.
TTrue
FFalse
Answer: False
This is the transposition fallacy (also called the prosecutor's fallacy). P(E|H) and P(H|E) are related by Bayes' theorem but are generally very different quantities. For example: the probability of testing positive given you have a disease might be 99%, but the probability of having the disease given a positive test might only be 50% if the disease is rare. Confusing these two probabilities is one of the most common reasoning errors in medicine, law, and everyday life.
Question 3 Short Answer
What is a 'prior probability' in Bayesian reasoning, and why does it matter when evaluating evidence?
Think about your answer, then reveal below.
Model answer: A prior probability is your initial estimate of how likely a hypothesis is before seeing a specific piece of evidence. It matters because the same evidence should update your belief by different amounts depending on how plausible the hypothesis was to begin with — extraordinary claims require extraordinary evidence because their prior is very low.
Bayesian updating says: posterior = prior × likelihood of evidence / normalizing factor. If the prior is very low (the hypothesis was already unlikely), even strong evidence may not make it probable. This explains why scientists require more evidence to overturn well-established theories than to confirm predictions of existing ones — the prior for radical claims is lower.