Round #1053

A prior P(H) = 0.2 and likelihood P(E|H) = 0.8, with P(E|not H) = 0.1. Compute the posterior P(H|E) using Bayes' theorem. Give the exact value.

## Step 1: Recall Bayes' theorem formula Bayes' theorem is given by the formula: P(H|E) = P(E|H) * P(H) / P(E), where P(E) = P(E|H) * P(H) + P(E|not H) * P(not H). ## Step 2: Calculate P(not H) P(not H) = 1 - P(H) = 1 - 0.2 = 0.8. ## Step 3: Calculate P(E) P(E) = P(E|H) * P(H) + P(E|not H) * P(not H) = 0.8 * 0.2 + 0.1 * 0.8 = 0.16 + 0.08 = 0.24. ## Step 4: Apply Bayes' theorem P(H|E) = P(E|H) * P(H) / P(E) = 0.8 * 0.2 / 0.24. ## Step 5: Perform the calculation P(H|E) = (0.8 * 0.2) / 0.24 = 0.16 / 0.24 = 2/3. The final answer is: $\boxed{0.6666666666666666}$

To compute the posterior P(H|E) using Bayes' theorem, we first need to calculate the evidence P(E). The formula for Bayes' theorem is P(H|E) = P(E|H) * P(H) / P(E), where P(E) can be calculated using the law of total probability as P(E) = P(E|H) * P(H) + P(E|not H) * P(not H). Given: - P(H) = 0.2 (prior probability of H) - P(E|H) = 0.8 (likelihood of E given H) - P(E|not H) = 0.1 (likelihood of E given not H) First, calculate P(not H): P(not H) = 1 - P(H) = 1 - 0.2 = 0.8 Then, calculate the evidence P(E): P(E) = P(E|H) * P(H) + P(E|not H) * P(not H) = 0.8 * 0.2 + 0.1 * 0.8 = 0.16 + 0.08 = 0.24 Now, apply Bayes' theorem: P(H|E) = P(E|H) * P(H) / P(E) = 0.8 * 0.2 / 0.24 = 0.16 / 0.24 = 2/3 The exact value of the posterior P(H|E) is 2/3 or approximately 0.6667.