## Step-by-Step Solution ### Step 1: Understand Bayes' Theorem Bayes' theorem is a fundamental rule in probability theory. It describes how to **update the probability of a hypothesis given new evidence**. ### Step 2: Recall the Formula Bayes' theorem states: \[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \] Where: - \( P(A|B) \): Probability of event A given B (posterior) - \( P(B|A) \): Probability of event B given A (likelihood) - \( P(A) \): Probability of event A (prior) - \( P(B) \): Probability of event B (marginal) ### Step 3: Analyze Each Option - **A) Compute joint probability:** Joint probability is calculated as \( P(A \cap B) \), but Bayes' theorem is primarily for conditional probabilities, not direct joint probability. - **B) Update probabilities given new evidence:** This is exactly what Bayes' theorem is for—it updates the probability of a hypothesis as new evidence (data) is observed. - **C) Find variance:** Variance is a measure of spread; Bayes' theorem does not compute variance. - **D) Calculate sample mean:** Sample mean is an average; Bayes' theorem is unrelated. ### Step 4: Select the Correct Answer The correct answer is: **B) Update probabilities given new evidence** --- ## Final Answer **Bayes' theorem is used to:** **B) Update probabilities given new evidence**