A doctor knows that 90% of patients have flu and 10%

Solution

Let the events be:

• F: patient has flu → P(F) = 0.90
• A: patient has allergy → P(A) = 0.10
• R: patient shows rash

Conditional probabilities:

• P(R | F) = 0.08
• P(R | A) = 0.95

We are asked to find: P(A | R) = Probability that a patient has allergy given that they have rash Apply Bayes’ Theorem: P(A | R) = [P(A) × P(R | A)] / P(R) Where: P(R) = P(F) × P(R | F) + P(A) × P(R | A) = (0.90 × 0.08) + (0.10 × 0.95) = 0.072 + 0.095 = 0.167 Now plug into Bayes' formula: P(A | R) = (0.10 × 0.95) / 0.167 = 0.095 / 0.167 ≈ 0.568