gen-define-bayes-theorem-and-give-one-short-worked-example-3763

Define Bayes' theorem and give one short worked example.

# Bayes' Theorem **Bayes' theorem** describes how to update the probability of a hypothesis based on new evidence: $$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$ Where: - **P(A|B)** = probability of A given B (posterior) - **P(B|A)** = probability of B given A (likelihood) - **P(A)** = probability of A (prior) - **P(B)** = probability of B (marginal) ## Worked Example **Problem:** A disease affects 1% of the population. A test is 90% accurate (correctly identifies both positive and negative cases). You test positive. What's the probability you have the disease? **Given:** - P(Disease) = 0.01 - P(Positive|Disease) = 0.90 - P(Positive|No Disease) = 0.10 **Solution:** First, find P(Positive): $$P(Positive) = P(Pos|Dis) \cdot P(Dis) + P(Pos|No\,Dis) \cdot P(No\,Dis)$$ $$= 0.90(0.01) + 0.10(0.99) = 0.009 + 0.099 = 0.108$$ Apply Bayes' theorem: $$P(Disease|Positive) = \frac{0.90 \times 0.01}{0.108} = \frac{0.009}{0.108} ≈ 0.083$$ **Answer:** Only **8.3%** chance you have the disease, despite testing positive! This shows why rare diseases need highly specific tests.