Bayes' Theorem Explained

• The Bayes' theorem describes the probability of an event based on prior knowledge of conditions related to the event.
• The Bayes' theorem is expressed as: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$, where:
  • $P(A|B)$ is the posterior probability of A given B is true.
  • $P(B|A)$ is the probability of B given A is true.
  • $P(A)$ is the prior probability of A.
  • $P(B)$ is the prior probability of B.

Derivation of Bayes' Theorem

• Bayes' theorem is derived from the definition of conditional probability.
• $P(A|B) = \frac{P(A \cap B)}{P(B)}$ and $P(B|A) = \frac{P(B \cap A)}{P(A)}$.
• Solving for $P(A \cap B)$ in both equations gives $P(A|B)P(B) = P(A \cap B) = P(B|A)P(A)$.
• Therefore, $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$.

Example Scenario

• A doctor tests a patient for a rare disease.
  • The disease affects 1 in 1,000 people.
  • The test has a 5% false positive rate.
  • The test has a 99% true positive rate.

Variables

• Define "Doença" as the patient has the disease, and "Positivo" as the patient tests positive.

Bayes' Theorem Applied

• $P(Doença|Positivo) = \frac{P(Positivo|Doença)P(Doença)}{P(Positivo)}$.
  • $P(Doença) = 1/1000 = 0.001$, which is the prior probability of having the disease.
  • $P(Positivo|Doença) = 0.99$, is the probability of testing positive if the patient has the disease.
  • $P(Positivo)$ is calculated using the law of total probability.

Law of Total Probability Application

• $P(Positivo) = P(Positivo|Doença)P(Doença) + P(Positivo|¬Doença)P(¬Doença)$.
  • $P(¬Doença) = 1 - P(Doença) = 0.999$.
  • $P(Positivo|¬Doença) = 0.05$, which is the probability of testing positive without the disease.
• $P(Positivo) = (0.99 \times 0.001) + (0.05 \times 0.999) = 0.00099 + 0.04995 = 0.05094$

Substitute Into Bayes' Theorem

• $P(Doença|Positivo) = \frac{0.99 \times 0.001}{0.05094} = \frac{0.00099}{0.05094} \approx 0.01943$
• Thus, the probability of truly having the disease, given a positive test, is approximately 1.94%.