Round #1083

A prior P(H) = 0.3 and likelihood P(E|H) = 0.9, with P(E|not H) = 0.2. 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 - 0.3 = 0.7. ## Step 3: Calculate P(E) P(E) = P(E|H) * P(H) + P(E|not H) * P(not H) = 0.9 * 0.3 + 0.2 * 0.7 = 0.27 + 0.14 = 0.41. ## Step 4: Apply Bayes' theorem P(H|E) = P(E|H) * P(H) / P(E) = 0.9 * 0.3 / 0.41. ## Step 5: Perform the calculation P(H|E) = 0.27 / 0.41 = 0.6585365853658537, rounding this to a more precise fraction gives us 0.6585365853658537, but for the sake of the format, we will keep it as a decimal and then convert it to a fraction for the final answer. The final answer is: $\boxed{0.6585365853658537}$

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) = 0.3 (prior probability of H) - P(E|H) = 0.9 (likelihood of E given H) - P(E|not H) = 0.2 (likelihood of E given not H) First, calculate P(not H): P(not H) = 1 - P(H) = 1 - 0.3 = 0.7 Then, calculate the evidence P(E): P(E) = P(E|H) * P(H) + P(E|not H) * P(not H) = 0.9 * 0.3 + 0.2 * 0.7 = 0.27 + 0.14 = 0.41 Now, apply Bayes' theorem to find P(H|E): P(H|E) = P(E|H) * P(H) / P(E) = 0.9 * 0.3 / 0.41 = 0.27 / 0.41 = 0.6585365853658537 Rounded to a more readable form, the posterior probability P(H|E) is approximately 0.659. ```csharp using System; class BayesTheoremCalculator { public static double CalculatePosterior(double priorH, double likelihoodEH, double likelihoodENotH) { // Calculate P(not H) double notH = 1 - priorH; // Calculate the evidence P(E) double evidenceE = likelihoodEH * priorH + likelihoodENotH * notH; // Apply Bayes' theorem to find P(H|E) double posteriorHE = (likelihoodEH * priorH) / evidenceE; return posteriorHE; } public static void Main(string[] args) { double priorH = 0.3; double likelihoodEH = 0.9; double likelihoodENotH = 0.2; double posteriorHE = CalculatePosterior(priorH, likelihoodEH, likelihoodENotH); Console.WriteLine("The posterior probability P(H|E) is: " + posteriorHE); } } ```